INTERNATIONAL GONGRESS

483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"> d = \gcd â¡ (q, r)

. For each such

m

, a straightforward application of the Chinese Remainder Theorem gives that the number of admissible pairs

(a, b)

⩽ d \prod_{p ∣ \gcd (d, q_{1} r_{1} m)} {(1 - \frac{1}{p})}_{p ∣ d, p ∤ q_{1} r_{1} m} (1 - \frac{2}{p}) ⩽ d \frac{\prod_{p ∣ d} {(1 - \frac{1}{p})}^{2}}{\prod_{p ∣ \gcd (d, q_{1} r_{1})} (1 - \frac{1}{p})} \cdot \frac{| m |}{φ (| m |)}

where we used that

1 - 2 / p ⩽ (1 - 1 / p)^{2}

. We then sum this inequality over

m

and use Lemma 3.4 below to complete the proof. (For full details, see [22] or [14, LEMMA 2.8].)

Lemma 3.4. Fix

C ⩾ 1

, and let

{(a_{p})}_{p}

prime be a sequence taking values in

[0, C]

. Then

\sum_{n ⩽ x} \prod_{p ∣ n} a_{p} ≪_{C} x \exp (\sum_{p ⩽ x} \frac{a_{p} - 1}{p}) for all x ⩾ 1

Proof. See Theorem 14.2 in [19].

3.3. Generalizing the ErdÃ¶s-Vaaler argument

The next step is to study averages of

\exp (\sum_{p ∣ q r / gdd (q, r)^{2}, p > M (q, r)} 1 / p)

. This gets a bit too technical in general, so we focus on the following special case:

Theorem 3.5. Let

Q ⩾ N ⩾ 2

, and let

S \subseteq {Q ⩽ q ⩽ 2 Q : q

square-free

}

be such that

\begin{matrix} (3.4) & N / 2 ⩽ \sum_{q \in S} \frac{φ (q)}{q} ⩽ N \end{matrix}

We then have

\begin{matrix} (3.5) & \sum_{q, r \in S} \frac{φ (q) φ (r)}{q r} \exp (\sum_{\begin{matrix} p ∣ q r / \gcd (q, r)^{2} \\ p > Q / [N \gcd (q, r)] \end{matrix}} \frac{1}{p}) ≪ N^{2} \end{matrix}

In particular, if

A_{q}^{*}

is as in (2.5) with

Δ_{q} = 1 / (q N)

, then meas

(⋃_{q \in S} A_{q}^{*}) ≫ 1

Remark. To see the last assertion, recall the notation

M (q, r) = 2 max {Δ_{q}, Δ_{r}} lcm [q, r]

from Lemma 3.3. By the assumptions of the theorem, we have

M (q, r) ⩾ 2 Q / [N \gcd (q, r)]

for

q, r \in S

. Hence, if (3.5) holds, then

\sum_{q, r \in S}

meas

(A_{q}^{*} \cap A_{r}^{*}) ≪ 1

by Lemma 3.3. We may then apply Proposition 3.2 to deduce that

meas (⋃_{q \in S} A_{q}^{*}) ≫ 1

When

N ≫ Q

, Theorem 3.5 follows from the work of ErdÅ‘Ìs and Vaaler (Theorem 2.12), but when

N = o (Q)

it was not known prior to [20] in this generality. The proof begins by adapting the ErdÅ‘s-Vaaler argument to this more general setup.

First, we must control the sum over primes in (3.5). Using (2.10) turns out to be too crude, so we modify our approach. Let

t_{j} = \exp (2^{j})

and

j_{0}

be such that

\sum_{t < p ⩽ t^{2}} 1 / p ⩽ 1

for

t ⩾ t_{j_{0}}

(

j_{0}

exists by Mertens' theorems [19, THEOREM 3.4].) Moreover, let

L (q, r) = \sum_{\begin{matrix} p ∣ q r / gdd (q, r)^{2} \\ p > Q / [N \gcd (q, r)] \end{matrix}} \frac{1}{p}, λ_{t} (q) = \sum_{\begin{matrix} p ∣ q \\ p > t \end{matrix}} \frac{1}{p}, L_{t} (q, r) = \sum_{\begin{matrix} p ∣ q r / \gcd (q, r)^{2} \\ p > t \end{matrix}} \frac{1}{p}

L_{t_{j_{0}}} (q, r) ⩽ 101

, then obviously

L (q, r) ≪ 1

. Otherwise, there is an integer

j ⩾ j_{0}

such that

L_{t_{j}} (q, r) > 101 ⩾ L_{t_{j + 1}} (q, r)

. Since

j ⩾ j_{0}

, we then also have

L_{t_{j + 1}} (q, r) > 100

. Now, note that if

Q / [N \gcd (q, r)] ⩾ t_{j + 1}

, then

L (q, r) ⩽ L_{t_{j + 1}} (q, r) ⩽ 101

To sum up,

L (q, r) ≪ 1

, unless

(q, r) \in B_{t_{j + 1}}

for some

j ⩾ j_{0}

, where

B_{t} := {(q, r) \in S \times S : \gcd (q, r) > Q / (N t), L_{t} (q, r) > 100}

We study the contribution of such pairs to the left-hand side of (3.5): if

(q, r) \in B_{t_{j + 1}}

, then

L (q, r) ⩽ 101 + \sum_{p ⩽ t_{j + 1}} \frac{1}{p} ⩽ \log \log t_{j + 1} + O (1) = j \log 2 + O (1)

by Mertens' estimate. In conclusion, Theorem 3.5 will follow if we can show that

\begin{matrix} (3.6) & \sum_{(q, r) \in B_{t}} \frac{φ (q) φ (r)}{q r} ≪ \frac{N^{2}}{t} for all t ⩾ t_{j_{0} + 1} \end{matrix}

Now, let us consider the special case when

N ≫ Q

, which corresponds to the ErdÅ‘s-Vaaler theorem. The inequality

\gcd (q, r) > Q / (N t)

is then basically trivially, so we
must prove (3.6) by exploiting the condition

L_{t} (q, r) > 100

. Indeed, writing

d = \gcd (q, r)

q = d q_{1}

and

r = d r_{1}

, we find that

λ_{t} (q_{1}) > 50

λ_{t} (r_{1}) > 50

. By symmetry, we have

# B_{t} ⩽ 2 \sum_{d ⩽ 2 Q} # {r_{1} ⩽ 2 Q / d} \cdot # {q_{1} ⩽ 2 Q / d : λ_{t} (q_{1}) > 50}

The number of

r_{1}

's is of course

⩽ 2 Q / d

. Moreover, using Chernoff's inequality and Lemma 3.4 with

a_{p} = \exp (1_{p > t} \cdot t / p)

, we find that

# {q_{1} ⩽ 2 Q / d : λ_{t} (q_{1}) > 50} ⩽ \sum_{q_{1} ⩽ 2 Q / d} \exp (- 50 t + t λ_{t} (p)) ≪ e^{- 50 t} Q / d

Putting everything together, we conclude that

\begin{matrix} (3.7) & # B_{t} ≪ e^{- t} Q^{2} for all t ⩾ 1 \end{matrix}

In particular, (3.6) holds, thus proving Theorem 3.5 when

N ≫ Q

On the other hand, if

N = o (Q)

, the condition that

\gcd (q, r) > Q / (N t)

for all

(q, r) \in B_{t}

is nontrivial and we must understand it and exploit it to prove Theorem 3.5. Indeed, if we treat the weights

φ (q) / q

as roughly constant in (3.4), we see that

δ

contains about

N

integers from

[Q, 2 Q]

, i.e., it is a rather sparse set. On the other hand, if

t

is not too large, then (3.7) gives no savings compared to the trivial upper bound

# B_{t} ⩽ # S^{2} \approx N^{2}

Since the condition that

L_{t} (q, r) > 100

is insufficient, let us ignore it temporarily and focus on the condition that

\gcd (q, r) > Q / (N t)

for all

(q, r) \in B_{t}

. There is an obvious way in which this condition can be satisfied for many pairs

(q, r) \in S \times S

, namely if there is some fixed integer

d > Q / (N t)

that divides a large proportion of integers in

S

. Notice that the number of total multiples of

d

[Q, 2 Q]

is about

Q / d < t \cdot N

, which is compatible with (3.4). We thus arrive at the following key question:

Model Problem. Let

D ⩾ 1

and

δ \in (0, 1]

, and let

δ \subseteq [Q, 2 Q] \cap Z

be a set of

≫ δ Q / D

elements such that there are

⩾ δ # S^{2}

pairs

(q, r) \in S \times S

with

\gcd (q, r) > D

. Must there be an integer

d > D

that divides

≫ δ^{100} Q / D

elements of

δ

It turns out that the answer to the Model Problem as stated is negative. However, a technical variant of it is true, that takes into account the weights

φ (q) / q

in (3.4) and (3.6), and that is asymmetric in

q

and

r

. We shall explain this in the next section.

For now, let us assume that the Model Problem as stated has an affirmative answer, and let us see how this yields Theorem 3.5. Suppose (3.6) fails for some

t

. By the Model Problem, there must exist an integer

d > N / (Q t)

dividing

≫ t^{- 100} # S

members of

S

. We might then also expect that

# B_{t} ≫ t^{- 200} # {(d m, d n) \in B_{t} : m, n ⩾ 1}

. But note that if

(q, r) = (d m, d n) \in B_{t}

, then

m, n ⩽ 2 Q / d < 2 t N

and

q r / \gcd (q, r)^{2} = m n / \gcd (m, n)^{2}

. In particular,

L_{t} (m, n) > 100

, so the argument leading to (3.7) implies that the number of

(d m, d n) \in B_{t}

≪ e^{- t} t^{2} N^{2}

. Hence,

B_{t} ≪ e^{- t} t^{202} N^{2} ≪ N^{2} / t

, as needed.

3.4. An iterative compression algorithm

To attack the Model Problem, we view it as a question in graph theory: consider the graph

G

, with vertex set

S

and edge set

B_{t}

. If the edge density of

G

⩾ 1 / t

, must there exist a dense subgraph

G^{'}

all of whose vertices are divisible by an integer

> Q / (N t)

To locate this "structured" subgraph

G^{'}

, we use an iterative "compression" argument, roughly inspired by the papers of ErdÃ³s-Ko-Rado [11] and Dyson [9]. With each iteration, we pass to a smaller set of vertices, where we have additional information about which primes divide them. Of course, we must ensure that we end up with a sizeable graph. We do this by judiciously choosing the new graph at each step so that it has at least as many edges as what the qualitative parameters of the old graph might naively suggest. This way the new graph will have improved "structure" and "quality." When the algorithm terminates, we will end up with a fully structured subset of

S

, where we know that all large GCDs are due to a large fixed common factor. This will then allow us to exploit the condition that

L_{t} (q, r) > 100

for all edges

(q, r)

. Importantly, our algorithm will also control the set

B_{t}

in terms of the terminal edge set. Hence the savings from the condition

L_{t} (q, r) > 100

in the terminal graph will be transferred to

B_{t}

, establishing (3.6).

One technical complication is that the iterative algorithm necessitates to view

G

as a bipartite graph. In addition, it is important to use the weights

φ (q) / q

. We thus set

μ (V) = \sum_{v \in V} \frac{φ (v)}{v} for V \subset N; μ (E) = \sum_{(v, w) \in E} \frac{φ (v) φ (w)}{v w} for E \subset N^{2}

Let us now explain the algorithm in more detail. We set

V_{0} = W_{0} = ς

and construct two decreasing sequences of sets

V_{0} \supseteq V_{1} \supseteq V_{2} \supseteq \dots

and

W_{0} \supseteq W_{1} \supseteq W_{2} \supseteq \dots

, as well as a sequence of distinct primes

p_{1}, p_{2}, \dots

such that either

p_{j}

divides all elements of

V_{j}

, or it is coprime to all elements of

V_{j}

(and similarly with

W_{j}

). Since

S

consists solely of square-free integers, there are integers

a_{j}, b_{j}

dividing

p_{1} \dots p_{j}

, and such that

\gcd (v, p_{1} \dots p_{j}) = a_{j}

and

\gcd (w, p_{1} \dots p_{j}) = b_{j}

for all

v \in V_{j}

and all

w \in W_{j}

Assume we have constructed

V_{i}, W_{i}, p_{i}

as above for

i = 1, \dots, j

. Let

ε_{i} =

B_{t} \cap (V_{i} \times W_{i})

be the edge sets. We then pick a new prime

p_{j + 1}

that occurs as common factor of

\gcd (v, w)

for at least one edge

(v, w) \in E_{j}

. (If there is no such prime, the algorithm terminates.) Then, we pick

V_{j + 1}

to be either

V_{j}^{(1)} := {v \in V_{j} : p_{j + 1} ∣ v}

V_{j}^{(0)} := {v \in V_{j}

p_{j + 1} ∤ v}

(and similarly with

W_{j + 1}

). Deciding how to make this choice is the most crucial part of the proof and we will analyze it in more detail below. At any rate, it is clear that after a finite number of steps, this process will terminate. We will thus arrive at sets of vertices

V_{J}

and

W_{J}

where

a = a_{J}

divides all elements of

V_{J}, b = b_{J}

divides all elements of

W_{J}

, and

\gcd (v, w) = \gcd (a, b)

for all edges

(v, w) \in E_{J} \subseteq B_{t}

. In particular,

\gcd (a, b) > Q / (N t)

, as long as

ε_{J} \neq \emptyset

. We have thus found our fixed large common divisor, so that we can use the ErdÅ‘s-Vaaler argument as in Section 3.3 to control the size of

E_{J}

. If we can ensure that

E_{J}

is a large enough portion of

E_{0} = B_{t}

, we will have completed the proof.

Let us now explain how to make the choice of which subgraph to focus on each time. Let

G_{j} = (V_{j}, W_{j}, E_{j})

be the bipartite graph at the

j

th iteration. Because we will use an unbounded number of iterations, it is important to ensure that

G_{j + 1}

has more edges than "what the qualitative parameters of

G_{j}

would typically predict." One way to assign meaning to this vague phrase is to use the edge density

# E_{j} / (# V_{j} # W_{j})

. Actually, in our case, we
should use the weighted density

δ_{j} = \frac{μ (E_{j})}{μ (V_{j}) μ (W_{j})}

Naively, we might guess that

μ (E_{j + 1}) \approx δ_{j} μ (V_{j + 1}) μ (W_{j + 1})

, meaning that

δ_{j + 1} \approx δ_{j}

. So we might try to choose

G_{j + 1}

so that

δ_{j + 1} ⩾ δ_{j}

. This would be analogous to Roth's "density increment" strategy [23,24]. Unfortunately, such an argument loses all control over the size of

E_{j}

, so we cannot use information on

E_{J}

to control

E_{0} = B_{t}

(which is our end goal).

In a completely different direction, we can use the special GCD structure of our graphs to come up with another "measure of quality" of our new graph compared to the old one. Recall the integers

a_{j + 1}

and

b_{j + 1}

. We then have

\begin{matrix} (3.8) & # E_{j} ⩽ # {m ⩽ \frac{2 Q}{a_{j}}, n ⩽ \frac{2 Q}{b_{j}} : \gcd (m, n) > \frac{Q / (N t)}{\gcd (a_{j}, b_{j})}} \end{matrix}

If all pairs

(m, n)

on the right-hand side of (3.8) were due to a fixed divisor of size

> [Q / (N t)] / \gcd (a_{j}, b_{j})

, then we would conclude that

# ε_{j} ≪ t^{2} N^{2} \cdot \frac{\gcd {(a_{j}, b_{j})}^{2}}{a_{j} b_{j}}

Actually, Green and Walker [13] proved recently that this bound is true, even without the presence of a universal divisor. So it makes sense to consider the quantity

# ε_{j} \cdot a_{j} b_{j} / \gcd {(a_{j}, b_{j})}^{2}

as a qualitative measure of

G_{j}

. As a matter of fact, since we are weighing

v

with

φ (v) / v

, and we have

φ (v) / v ⩽ φ (a_{j}) / a_{j}

whenever

a_{j} ∣ v

, we may even consider

λ_{j} := \frac{a_{j} b_{j}}{\gcd {(a_{j}, b_{j})}^{2}} \cdot \frac{a_{j} b_{j}}{φ (a_{j}) φ (b_{j})} \cdot μ (ε_{j})

Let us see a different argument for why this quantity might be a good choice, by studying the effect of each prime

p \in {p_{1}, \dots, p_{j}}

to the parameters

Q / a_{j}, Q / b_{j}

and

[Q / (N t)] / \gcd (a_{j}, b_{j})

that control the size of

m, n

, and

\gcd (m, n)

, respectively, in (3.8):

Case 1: $p ∣ a_{j}$ and $p ∣ b_{j}$ . Then $p$ reduces the upper bounds on the size of both $m$ and $n$ by a factor $1 / p$ . On the other hand, it also reduces the lower bound on their GCD (that affects both $m$ and $n$ ) by $1 / p$ . Hence, we are in a balanced situation.
Case 2: $p ∤ a_{j}$ and $p ∤ b_{j}$ . In this case, $p$ affects no parameters.
Case 3: $p ∣ a_{j}$ and $p ∤ b_{j}$ . Then $p$ reduces the upper bound on $m$ by a factor $1 / p$ , but it does not affect the bound on $n$ nor on $\gcd (m, n)$ . This is an advantageous situation, gaining us a factor of $p$ compared to what we had. Accordingly, $λ_{j}$ is multiplied by $p$ in this case. This gain allows us to afford a big loss of vertices when falling in this "asymmetric" case (a proportion of $1 - O (1 / p)$ ).
Case 4: $p ∤ a_{j}$ and $p ∣ b_{j}$ . Then we gain a factor of $p$ as in the previous case.

Iteratively increasing

λ_{j}

would be adequate for showing (3.6), by mimicking the ErdÅ‘s-Vaaler argument from Section 3.3. Unfortunately, it is not possible to guarantee that

λ_{j}

increases at each stage because it is not sensitive enough to the edge density, and so this proposal also fails. However, we will show that (a small variation of) the hybrid quantity

\begin{matrix} (3.9) & q_{j} := δ_{j}^{9} λ_{j} \end{matrix}

can be made to increase at each step, while keeping track of the sizes of the vertex sets. We call

q_{j}

the quality of

G_{j}

Let us now discuss how we might carry out the "quality increment" strategy. Given

V_{j}

and

p_{j + 1}

, we wish to set

V_{j + 1} = V_{j}^{(k)}

and

W_{j + 1} = W_{j}^{(ℓ)}

for some

k, ℓ \in {0, 1}

. Let us call

G_{j}^{(k, ℓ)}

each of the four potential choices for

G_{j + 1}

. For their quality

q_{j}^{(k, ℓ)}

, we have:

\frac{q_{j}^{(1, 1)}}{q_{j}} = {(\frac{δ^{(1, 1)}}{δ_{j}})}^{10} α β {(1 - \frac{1}{p})}^{- 2}, \frac{q_{j}^{(1, 0)}}{q_{j}} = {(\frac{δ_{j}^{(1, 0)}}{δ_{j}})}^{10} α (1 - β) p {(1 - \frac{1}{p})}^{- 1}

\frac{q_{j}^{(0, 1)}}{q_{j}} = {(\frac{δ_{j}^{(0, 1)}}{δ_{j}})}^{10} (1 - α) β p {(1 - \frac{1}{p})}^{- 1}, \frac{q_{j}^{(0, 0)}}{q_{j}} = {(\frac{δ_{j}^{(0, 0)}}{δ_{j}})}^{10} (1 - α) (1 - β)

where

δ_{j}^{(k, ℓ)}

is the edge density of

G_{j}^{(k, ℓ)}, α = μ (V_{j}^{(1)}) / μ (V_{j})

is the proportion of vertices in

V_{j}

that are divisible by

p_{j + 1}

, and similarly

β = μ (W_{j}^{(1)}) / μ (W_{j})

. In addition, we have

δ_{j}^{(1, 1)} α β + δ_{j}^{(1, 0)} α (1 - β) + δ_{j}^{(0, 1)} (1 - α) β + δ_{j}^{(0, 0)} (1 - α) (1 - β) = δ_{j}

so that if one of the

δ_{j}^{(k, ℓ)}

, s is smaller than

δ

, some other must be larger. Such an unbalanced situation is advantageous, so let us assume that

δ_{j}^{(k, ℓ)} \sim δ_{j}

for all

k, ℓ

Notice that we have an extra factor

p

in the asymmetric cases

(0, 1)

and

(1, 0)

. We can then easily obtain a quality increment in one of these two cases, unless

α, β ≪ 1 / p

, or if

α, β ⩾ 1 - O (1 / p)

. It turns out that the former case can be treated with a trick. So, the real difficulty is to obtain a quality increment when

α, β

are both close to 1 . As a matter of fact, the critical case is when

α, β \sim 1 - 1 / p

. Indeed, we then have

q_{j}^{(k, ℓ)} \sim 1

in all four cases, so we maintain a constant quality no matter which subgraph we choose to focus on.

It is important to remark here that the factor

(1 - 1 / p)^{- 2}

in the case

(k, ℓ) = (1, 1)

is essential (the factors

(1 - 1 / p)^{- 1}

in the asymmetric cases are less important as it turns out). Without this extra factor, we would not have been able to guarantee that the quality stays at least as big as

q_{j}

. Crucially, this factor originates from the weights

φ (v) / v

of the vertices that are naturally built in the Duffin-Schaeffer conjecture and that dampen down contributions from integers with too many prime divisors.

We conclude this discussion by going back to the Model Problem. We mentioned in Section 3.3 that this problem is false. The reason is a counterexample due to Sam Chow, a square-free version of which is given by

S = {P / j : j ∣ P, x / 2 ⩽ j ⩽ x}

with

P =

\prod_{p ⩽ x} p

. Indeed, all pairwise GCDs here are

⩾ P / x^{2}

, but there is no fixed integer of size

≫ P / x^{2}

dividing a positive proportion of elements of this set. In addition, note that if

p ⩽

x / \log x

, then the proportion of

S

divisible by

p

\sim 1 - 1 / p

, just like in the critical case discussed above.

3.5. The quality increment argument

We now discuss the formal details of our iterative algorithm. We must first set up some notation. We say that

G = (V, W, E, P, a, b)

is a square-free

G C D

graph if:

$V$ and $W$ are nonempty, finite sets of square-free integers;
$(V, W, E)$ is a bipartite graph, meaning that $E \subseteq V \times W$ ;
$P$ is a finite set of primes, and $a$ and $b$ divide $\prod_{p \in P} p$ ;
$a ∣ v$ and $b ∣ w$ for all $(v, w) \in V \times W$ ;
if $(v, w) \in E$ and $p \in P$ , then $p ∣ \gcd (v, w)$ precisely when $p ∣ \gcd (a, b)$ .

We shall refer to

(P, a, b)

as the multiplicative data of

G

. Furthermore, we defined the edge density of

G

δ (G) := \frac{μ (E)}{μ (V) μ (W)}

, and its quality by

q (G) := δ (G)^{9} \cdot μ (E) \cdot \frac{a b}{\gcd (a, b)^{2}} \cdot \frac{a b}{φ (a b)} \cdot \prod_{p \in P} {(1 - \frac{1}{p^{3 / 2}})}^{- 10}

In addition, we define the set of "remaining large primes" of

G

R (G) := {p \notin P : p > 5^{100}, p ∣ \gcd (v, w) for some (v, w) \in E}

Finally, if

G^{'} = (V^{'}, W^{'}, E^{'}, P^{'}, a^{'}, b^{'})

is another square-free GCD graph, we call it a subgraph of

G

V^{'} \subseteq V, W^{'} \subseteq W, E^{'} \subseteq E, P^{'} \supseteq P, \prod_{p ∣ a^{'}, p \in P} p = a, \prod_{p ∣ b^{'}, p \in P} p = b

Lemma 3.6 (The quality increment argument). Let

G = (V, W, E, P, a, b)

be a square-free GCD graph, let

p \in R (G)

, and let

α = \frac{μ ({v \in V : p ∣ v})}{μ (V)}

and

β = \frac{μ ({w \in W : p ∣ w})}{μ (W)}

(a) If

min {α, β} ⩽ 1 - 5^{12} / p

, then there is a subgraph

G^{'}

G

with multiplicative data

(P \cup {p}, a p^{k}, b p^{ℓ})

for some

k, ℓ \in {0, 1}

satisfying

δ {(G^{'})}^{m} q (G^{'}) ⩾

2^{k \neq ℓ} δ (G)^{m} q (G)

for

m \in {0, 1}

(b) If

min {α, β} > 1 - 5^{12} / p

, then there is a subgraph

G^{'}

G

with set of primes

P \cup {p}

and with quality

q (G^{'}) ⩾ q (G)

Proof. Each pair of

k, ℓ \in {0, 1}

defines a subgraph of

G

with multiplicative data

(P \cup {p}, a p^{k}, b p^{ℓ})

. Indeed, we merely need to focus on the vertex subsets

V_{k} = {v \in

V : p^{k} ∥ v}

and

W_{ℓ} = {w \in W : p^{ℓ} ∥ w}

. This new GCD graph is formally given by the sextuple

G_{k, ℓ} = (V_{k}, W_{ℓ}, E_{k, ℓ}, P \cup {p}, a p^{k}, b p^{ℓ})

, where

E_{k, ℓ} = E \cap (V_{k} \times W_{ℓ})

. Let

δ_{k, ℓ} = \frac{μ (E_{k, ℓ})}{μ (E)}, α_{k} = \frac{μ (V_{k})}{μ (V)}

and

β_{ℓ} = \frac{μ (W_{ℓ})}{μ (W)}

, so that

α_{0} = 1 - α, α_{1} = α, β_{0} = 1 - β

and

β_{1} = β

. We then have

\begin{matrix} (3.10) & \frac{δ {(G_{k, ℓ})}^{m} q (G_{k, ℓ})}{δ (G)^{m} q (G)} = \frac{δ_{k, ℓ}^{10 + m} {(α_{k} β_{ℓ})}^{- 9 - m} p^{1} k \neq ℓ}{(1 - 1 / p)^{k + ℓ} {(1 - p^{- 3 / 2})}^{10}} \end{matrix}

(a) We claim that there exist choices of

k, ℓ \in {0, 1}

such that

\begin{matrix} (3.11) & δ_{k, ℓ} ⩾ {\begin{cases} {(α_{k} β_{k})}^{9 / 10} & if k = ℓ \\ \frac{α (1 - β) + (1 - α) β}{5} & if k \neq ℓ \end{cases} \end{matrix}

To prove this claim, it suffices to show that

\begin{matrix} (3.12) & (α β)^{9 / 10} + ((1 - α) (1 - β))^{9 / 10} + \frac{2}{5} \cdot [α (1 - β) + (1 - α) β] ⩽ 1 \end{matrix}

Let

u = max {α β, (1 - α) (1 - β)}

. Then

(α β)^{9 / 10} + ((1 - α) (1 - β))^{9 / 10} ⩽ u^{2 / 5} [(α β)^{1 / 2} + ((1 - α) (1 - β))^{1 / 2}] ⩽ u^{2 / 5}

by the Cauchy-Schwarz inequality. On the other hand, we have

α (1 - β) + (1 - α) β = 1 - α β - (1 - α) (1 - β) ⩽ 1 - u

In conclusion, the left-hand side of

(3.12)

⩽ u^{2 / 5} + 2 (1 - u) / 5 ⩽ 1

, as needed.

Now, if (3.11) is true with

k = ℓ

, part (a) of the lemma follows immediately by (3.10) upon taking

G^{'} = G_{k, k}

. Assume then that (3.11) fails when

k = ℓ

. We separate two cases.

Case 1:

max {α, β} > 5^{12} / p

. We know that (3.11) holds for some choice of

k \neq ℓ

. Suppose that

k = 1

and

ℓ = 0

for the sake of concreteness; the other case is similar. Then, (3.10) implies

\begin{aligned} \frac{δ {(G_{1, 0})}^{m} q (G_{1, 0})}{δ (G)^{m} q (G)} & ⩾ {(\frac{α (1 - β) + β (1 - α)}{5})}^{10 + m} \frac{p}{(α (1 - β))^{9 + m}} \\ ⩾ \frac{α (1 - β) + β (1 - α)}{5^{11}} p \end{aligned}

for

m ⩽ 1

. The proof is complete by taking

G^{'} = G_{1, 0}

, unless

α (1 - β) + β (1 - α) <

2 \cdot 5^{11} / p

. In this case, we claim that either

max {α, β} < 5^{12} / p

min {α, β} > 1 - 5^{12} / p

(both of which we have assumed are false). By symmetry, we may assume

α ⩽ 1 / 2

. Then

β / 2 ⩽ β (1 - α) < 2 \cdot 5^{11} / p

, as needed. In particular,

β ⩽ 1 / 2

(because

p > 5^{100}

), and thus

α / 2 ⩽ α (1 - β) < 2 \cdot 5^{11} / p

. We have thus reached a contradiction. This proves the lemma in this case.

Case 2:

max {α, β} ⩽ 5^{12} / p

. We must then have

δ_{1, 1} ⩽ (α β)^{9 / 10} ⩽ 5^{22} p^{- 9 / 5}

. Let us now define the GCD subgraph

G^{'} = (V, W, E^{'}, P \cup {p}, a, b)

, where

E^{'} = E ∖ (V_{1} \times W_{1})

. Notice that we trivially have

a ∣ v

and

b ∣ w

. In addition, since we have removed all edges

(v, w)

where

p

divides both

v

and

w

, we must have that

p ∤ \gcd (v, w)

whenever

(v, w) \in E^{'}

. So, indeed, we see that

G^{'}

is a GCD subgraph of

G

. Moreover,

\frac{δ {(G^{'})}^{m} q (G^{'})}{δ (G)^{m} q (G)} = {(\frac{μ (E^{'})}{μ (E)})}^{10 + m} {(1 - p^{- 3 / 2})}^{- 10} = {(1 - δ_{1, 1})}^{10 + m} {(1 - p^{- 3 / 2})}^{- 10} ⩾ 1

for

m ⩽ 1

, because

δ_{1, 1} ⩽ 5^{22} p^{- 9 / 5}

and

p > 5^{100}

. This proves the lemma in this case, too.

(b) Let

c = {(1 - p^{- 3 / 2})}^{- 1}

. Using (3.10), we get a quality increment by letting

G^{'} = G_{k, ℓ}

if one of the following inequalities holds:

\begin{matrix} (3.13) & \begin{array}{ll} c δ_{1, 1} ⩾ (α β)^{9 / 10} (1 - 1 / p)^{2 / 10}, & c δ_{0, 0} ⩾ ((1 - α) (1 - β))^{9 / 10} \\ c δ_{1, 0} ⩾ (α (1 - β))^{9 / 10} p^{- 1 / 10}, & c δ_{0, 1} ⩾ ((1 - α) β)^{9 / 10} p^{- 1 / 10} \end{array} \end{matrix}

Let

α = 1 - A / p

and

β = 1 - B / p

with

A, B \in [0, 5^{12}]

. It suffices to show that

\begin{matrix} (3.14) & c ⩾ {(1 - \frac{A}{p})}^{\frac{9}{10}} {(1 - \frac{B}{p})}^{\frac{9}{10}} {(1 - \frac{1}{p})}^{\frac{2}{10}} + \frac{(A B)^{\frac{9}{10}}}{p^{9 / 5}} + \frac{{(1 - \frac{A}{p})}^{\frac{9}{10}} B^{\frac{9}{10}} + A^{\frac{9}{10}} {(1 - \frac{B}{p})}^{\frac{9}{10}}}{p} \end{matrix}

Indeed, the right-hand side of (3.14) is

\begin{aligned} ⩽ \exp (- \frac{0.9 A + 0.9 B + 0.2}{p}) + \frac{5^{22}}{p^{9 / 5}} + \frac{A^{9 / 10} + B^{9 / 10}}{p} \\ ⩽ 1 - \frac{0 . A + 0.9 B + 0.2}{p} + \frac{(0.9 A + 0.9 B + 0.2)^{2}}{2 p^{2}} + \frac{5^{22}}{p^{9 / 5}} + \frac{A^{9 / 10} + B^{9 / 10}}{p} \end{aligned}

where we used the inequalities

0 ⩽ 1 - x ⩽ e^{- x} ⩽ 1 - x + x^{2} / 2

, valid for all

x \in [0, 1]

. By the inequality of arithmetic and geometric means, we have

0.9 A + 0.1 ⩾ A^{9 / 10}

and

0.9 B + 0.1 ⩾ B^{9 / 10}

. Hence, the right-hand side of (3.14) is

⩽ 1 + \frac{(0.9 A + 0.9 B + 0.2)^{2}}{2 p^{2}} + \frac{5^{11}}{p^{9 / 5}} ⩽ 1 + \frac{5^{25}}{p^{2}} + \frac{5^{22}}{p^{9 / 5}} ⩽ 1 + \frac{1}{p^{3 / 2}} ⩽ c

for

p > 5^{100}

. This completes the proof of the part (b) of the lemma.

3.6. Proof of Theorem 3.5

Let

Q, N, S

and

B_{t}

with

t ⩾ t_{j_{0} + 1}

be as in Section 3.3. We want to prove (3.6). We may assume that

μ (B_{t}) ⩾ μ (S)^{2} / t

; otherwise, (3.6) is trivially true.

Consider the GCD graph

G_{0} := (S, S, B_{t}, \emptyset, \emptyset, \emptyset)

, and note that

δ (G_{0}) ⩾ 1 / t

. We repeatedly apply part (a) of Lemma 3.6 to create a sequence of distinct primes

p_{1}, p_{2}, \dots

and of square-free GCD graphs

G_{j} = (V_{j}, W_{j}, ε_{j}, {p_{1}, \dots, p_{j}}, a_{j}, b_{j}), j = 1, 2, \dots

, with

G_{j}

a subgraph of

G_{j - 1}

. Assuming we have applied Lemma 3.6(a)

j

times, we may apply it once more if there is

p \in R (G_{j})

dividing a proportion

⩽ 1 - 5^{12} / p

V_{j}

and

W_{j}

Naturally, the above process will terminate after a finite time, say after

J_{1}

steps and we will arrive at a GCD graph

G_{J_{1}}

such that if

p \in R (G_{J_{1}})

, then

p

divides a proportion

> 1 - 5^{12} / p

of the vertex sets

V_{J_{1}}

and

W_{J_{1}}

. In addition, the sequence of GCD graphs produced is such that

δ {(G_{j})}^{m} q (G_{j}) ⩾ 2^{1_{j \in D}} δ {(G_{j - 1})}^{m} q (G^{(j - 1)})

for

m \in {0, 1}

, where

D = {j ⩽ J_{1} : p_{j} divides a_{J_{1}} b_{J_{1}} / \gcd {(a_{J_{1}}, b_{J_{1}})}^{2}}

In particular,

\begin{matrix} (3.15) & δ {(G_{J_{1}})}^{m} q (G_{J_{1}}) ⩾ 2^{# D} δ {(G_{0})}^{m} q (G_{0}) for m \in {0, 1} \end{matrix}

To proceed, we must separate two cases.

Case 1:

q (G_{J_{1}}) ⩾ t^{30} q (G_{0})

. We apply repeatedly Lemma 3.6 (either part (a) or (b), according to whether the condition

min {α, β} ⩽ 1 - 5^{12} / p

holds or fails) to create a sequence of primes

p_{J_{1} + 1}, p_{J_{1} + 2}, \dots

that are distinct from each other and from

p_{1}, \dots, p_{J_{1}}

, and of square-free GCD graphs

G_{j} = (V_{j}, W_{j}, E_{j}, {p_{1}, \dots, p_{j}}, a_{j}, b_{j}), j = J_{1} + j, J_{1} + 2, \dots

,
with

G_{j}

a subgraph of

G_{j - 1}

. As before, this process will terminate, say after

J_{2} - J_{1}

steps, and we will arrive at a GCD graph

G_{J_{2}}

with

R (G_{J_{2}}) = \emptyset

. By construction, we have

q (G_{J_{2}}) ⩾ q (G_{J_{2} - 1}) ⩾ \dots ⩾ q (G_{J_{1}}) ⩾ t^{30} q (G_{0})

In addition, we have

\begin{matrix} (3.16) & q (G_{0}) = \frac{μ {(B_{t})}^{10}}{μ (S)^{18}} ⩾ \frac{μ {(B_{t})}^{10}}{N^{18}} \end{matrix}

by (3.4). (In particular, note that

q (G_{0}) > 0

, so

q (G_{J_{2}}) > 0

and thus

E_{J_{2}} \neq \emptyset

.) On the other hand, if we let

a = a_{J_{2}}

and

b = b_{J_{2}}

, then

\gcd (v, w) ∣ \gcd (a, b) P

with

P = \prod_{p ⩽ 5^{100}} p

for all

(v, w) \in E_{J_{2}}

. In particular,

\gcd (a, b) > Q / (P N t)

. Moreover,

μ (E_{J_{2}}) ⩽ \sum_{m ⩽ 2 Q / a, n ⩽ 2 Q / b} \frac{φ (a m)}{a m} \cdot \frac{φ (b n)}{b n} ⩽ \frac{φ (a) φ (b)}{a b} \cdot \frac{4 Q^{2}}{a b}

Since

δ (G_{J_{2}}) ⩽ 1

and

\prod_{p} {(1 - 1 / p^{3 / 2})}^{- 10} < \infty

, we then have

\begin{matrix} (3.17) & q (G_{J_{2}}) ≪ μ (E_{J_{2}}) \frac{a b}{\gcd (a, b)^{2}} \frac{a b}{φ (a) φ (b)} ≪ t^{2} N^{2} \end{matrix}

Recalling that

q (G_{J_{2}}) ⩾ t^{30} q (G_{0})

, relations (3.16) and (3.17) complete the proof of (3.6), and thus of Theorem 3.5 in this case.

Case 2:

q (G_{J_{1}}) < t^{30} q (G_{0})

. In this case, we do not have such a big quality gain, so we need to use that

L_{t} (v, w) > 100

for all

(v, w) \in B_{t}

. But we must be very careful because this condition might be dominated by the prime divisors of the fixed integers

a

and

b

we are constructing. Before we proceed, note that (3.15) implies that

\begin{matrix} (3.18) & δ (G_{J_{1}}) ⩾ δ (G_{0}) \cdot \frac{q (G_{0})}{q (G_{J_{1}})} ⩾ \frac{1}{t} \cdot \frac{1}{t^{30}} = \frac{1}{t^{31}} \end{matrix}

Let

R = R (J_{1})

and let

p \in R

. By the construction of

G_{J_{1}}, p

divides a proportion

> 1 - 5^{12} / p

of the vertex sets

V_{J_{1}}

and

W_{J_{1}}

. Therefore,

μ ({(v, w) \in E_{J_{1}} : p ∣ v w / \gcd (v, w)^{2}}) ⩽ \frac{2 \cdot 5^{12}}{p} μ (V_{J_{1}}) μ (W_{J_{1}}) ⩽ \frac{5^{13} t^{31}}{p} μ (E_{J_{1}})

where we used (3.18). As a consequence, we find

\sum_{(v, w) \in E_{J_{1}}} \frac{φ (v) φ (w)}{v w} \sum_{\begin{matrix} p > >^{32}, p \in R \\ p ∣ v w / \gcd (v, w)^{2} \end{matrix}} \frac{1}{p} ⩽ \sum_{p > t^{32}} \frac{5^{13} t^{31} μ (E_{J_{1}})}{p^{2}} ⩽ \frac{μ (E_{J_{1}})}{100}

Hence, if we let

E_{J_{1}}^{g o o d} = {(v, w) \in E_{J_{1}} : \sum_{\begin{matrix} p > t^{32}, p \in R \\ p ∣ v w / \gcd (v, w)^{2} \end{matrix}} \frac{1}{p} ⩽ 1}

Markov's inequality implies that

μ (E_{J_{1}}^{good}) ⩾ 0.99 μ (E_{J_{1}})

. We then define the GCD graph

G_{0}^{'} = (V_{J_{1}}, W_{J_{1}}, E_{J_{1}}^{good}, P, a_{J_{1}}, b_{J_{1}})

. Note that

g (G_{0}^{'}) = {(\frac{μ (E_{J_{1}}^{g o o d})}{μ (E_{J_{1}})})}^{10} q (G_{J_{1}}) ⩾ \frac{q (G_{J_{1}})}{2} ⩾ \frac{q (G_{0})}{2}

Next, we apply repeatedly Lemma 3.6 to create a sequence of distinct primes

p_{J_{1} + 1}, p_{J_{1} + 2}, \dots \in R

and of GCD graphs

G_{j}^{'} = (V_{j}^{'}, W_{j}^{'}, E_{j}^{'}, {p_{1}, \dots, p_{J_{1} + j}}, a_{j}^{'}, b_{j}^{'})

j = 1, \dots

, with

G_{j}^{'}

a subgraph of

G_{j - 1}^{'}

. This process will terminate, say after

K

steps, and we will arrive at a GCD graph

G_{K}^{'}

with

R (G_{K}^{'}) = \emptyset

. By construction, we have

\begin{matrix} (3.19) & q (G_{K}^{'}) ⩾ q (G_{K - 1}^{'}) ⩾ \dots ⩾ q (G_{0}^{'}) ⩾ q (G_{0}) / 2 > 0 \end{matrix}

In particular,

E_{K}^{'} \neq \emptyset

. It remains to give an upper bound on

q (G_{K}^{'})

Let

a^{'} = a_{K}^{'}

and

b = b_{K}^{'}

, and recall that

P = \prod_{p ⩽ 5^{100}} p

. Then

\gcd (v, w) ∣ \gcd (a^{'}, b^{'}) P

for all

(v, w) \in E_{K}^{'}

. In particular,

\gcd (a^{'}, b^{'}) > Q / (P N t)

. Moreover, if

(v, w) \in E_{K}^{'}

and we let

v = a^{'} m

and

w = b^{'} n

, then

100 < L_{t} (v, w) ⩽ 5 + L_{t^{32}} (v, w) ⩽ 6 + \sum_{\begin{matrix} p > t^{32}, p \notin R \\ p ∣ v w / ccd (v, w)^{2} \end{matrix}} \frac{1}{p} ⩽ 6 + \frac{# D}{t^{32}} + L_{t^{32} (m, n)}

where the first inequality is true because

(v, w) \in B_{t}

, the second because

\sum_{y < p ⩽ y^{2}} 1 / p ⩽ 1

for

y ⩾ t_{j_{0}}

, the third because

(v, w) \in E_{J_{1}}^{good}

, and the fourth because if

p

divides

a^{'} b^{'} / \gcd {(a^{'}, b^{'})}^{2}

and

p \notin R

, then

p \in D

. Now, since

2^{# D} ⩽ q (G_{J_{1}}) / q (G_{0}) ⩽ t^{30}

, we have that

L_{t 32} (m, n) > 93

. Therefore,

μ (E_{K}^{'}) ⩽ \sum_{\begin{matrix} m ⩽ 2 Q / a^{'}, n ⩽ 2 Q / b^{'} \\ L_{t} 32 \\ (m, n) > 93 \end{matrix}} \frac{φ (a^{'}) φ (b^{'})}{a^{'} b^{'}} ≪ \frac{φ (a^{'}) φ (b^{'})}{a^{'} b^{'}} \cdot \frac{4 Q^{2}}{a^{'} b^{'}} e^{- t^{32}}

by arguing as in the proof of (3.7). We may then insert this inequality into the definition of

q (G_{K}^{'})

and conclude that

q (G_{K}^{'}) ≪ e^{- t^{32}} t^{2} N^{2}

. Together with (3.19) and (3.16), this completes the proof of (3.6), and thus of Theorem 3.5 in this last case as well.

ACKNOWLEDGMENTS

The author is grateful to James Maynard for his comments on a preliminary version of the paper.

FUNDING

The author is supported by the Courtois Chair II in fundamental research of the UniversitÃ© de MontrÃ©al, the Natural Sciences and Engineering Research Council of Canada (Discovery Grant 2018-05699), and the Fonds de recherche du QuÃ©bec - Nature et technologies (projets de recherche en Ã©quipe 256442 and 300951 ).

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DIMITRIS KOUKOULOPOULOS

DÃ©partement de mathÃ©matiques et de statistique, UniversitÃ© de MontrÃ©al, CP 6128 succ.

Centre-Ville, MontrÃ©al, QC H3C 3J7, Canada, dimitris.koukoulopoulos @ umontreal.ca

EULER SYSTEMS AND THE BLOCH-KATO CONJECTURE FOR AUTOMORPHIC GALOIS REPRESENTATIONS

DAVID LOEFFLER AND SARAH LIVIA ZERBES

Abstract

We survey recent progress on the Bloch-Kato conjecture, relating special values of $L$ functions to cohomology of Galois representations, via the machinery of Euler systems. This includes new techniques for the construction of Euler systems, via the Ã©tale cohomology of Shimura varieties, and new methods for proving explicit reciprocity laws, relating Euler systems to critical values of $L$ -functions. These techniques have recently been used to prove the Bloch-Kato conjecture for critical values of the degree $4 L$ -function of ${G S p}_{4}$ , and we survey ongoing work aiming to apply this result to the Birch-SwinnertonDyer conjecture for modular abelian surfaces, and to generalise it to a range of other automorphic $L$ -functions.

MATHEMATICS SUBJECT CLASSIFICATION 2020

Primary 11G40; Secondary 11F67, 11F80, 11G18

KEYWORDS

Bloch-Kato conjecture, Euler system, Selmer group

1. WHAT IS THE BLOCH-KATO CONJECTURE?

The Bloch-Kato conjecture, formulated in [11], relates the cohomology of global Galois representations to the special values of

L

-functions. We briefly recall a weak form of the conjecture, which will suffice for this survey. Let

L / Q_{p}

be a finite extension, let

K

be a number field, and let

V

be a representation of

Γ_{K} = Gal (\bar{K} / K)

on a finite-dimensional

L

-vector space. We suppose

V

is unramified outside finitely many primes and de Rham at the primes above

p

. Then we may attach to

V

the following two objects:

Its $L$ -function, which is the formal Euler product

L (V, s) = \prod_{v} P_{v} {(V, N (v)^{- s})}^{- 1}

where

v

varies over (finite) primes of

K

, and

P_{v} (V, X) \in L [X]

is a local Euler factor depending on the restriction of

V

to a decomposition group at

v

. It is conjectured that, for any choice of isomorphism

\bar{L} ≅ C

, this product converges for

ℜ (s) ≫ 0

and and has meromorphic continuation to all of

C

Its Selmer group $H_{f}^{1} (K, V)$ , a certain (finite-dimensional) subspace of the Galois cohomology group $H^{1} (K, V)$ determined by local conditions at each prime, defined in $[11]$ .

The weak Bloch-Kato conjecture asserts that

{ord}_{s = 1} L (V^{*}, s) = \dim H_{f}^{1} (K, V) - \dim H^{0} (K, V)

The full conjecture as formulated in [11] also determines the leading term of

L (V^{*}, s)

s = 1

up to a

p

-adic unit, in terms of the cohomology of an integral lattice

T \subset V

This conjecture includes as special cases a wide variety of well-known results and conjectures. For example, when

V

is the 1-dimensional trivial representation, the weak conjecture states that

ζ_{K} (s)

has a simple pole at

s = 1

; and the strong conjecture (for all

p

at once) is equivalent to the analytic class number formula, relating the residue at this pole to the class group and unit group of

K

. If

V = T_{p} (E) \otimes Q_{p}

, where

E

is an elliptic curve over

K

and

T_{p} (E)

is its Tate module, then

L (V^{*}, s)

is the Hasse-Weil

L

-function

L (E / K, s)

, and we recover the Birch and Swinnerton-Dyer conjecture for

E

over

K

Critical values

The

L

-function

L (V^{*}, s)

is expected to satisfy a functional equation relating

L (V, s)

and

L (V^{*}, 1 - s)

, after multiplying by a suitable product of

Γ

-functions

L_{\infty} (V^{*}, s)

(determined by the Hodge-Tate weights of

V

p

and the action of complex conjugation). These

Γ

-factors may have poles at

s = 1

, forcing

L (V^{*}, 1)

to vanish.

Following [49], we say

V

r

-critical, for some

r ⩾ 0

, if

L_{\infty} (V^{*}, 1 - s)

has a pole of order

r

s = 0

, and

L_{\infty} (V, s)

is holomorphic there. In particular,

V

is 0 -critical if

L (V^{*}, 1)

is a critical value in the sense of Deligne [18]. The most interesting cases of the Bloch-Kato conjecture are when

V

is 0 -critical, and it is these which our main theorems below will address; but 1-critical Galois representations will also play a crucial auxiliary role in our strategy.

Iwasawa theory

The Bloch-Kato conjecture is closely related to the Iwasawa main conjecture, in which the finite-dimensional Selmer group

H_{f}^{1} (K, V)

is replaced by a finitely-generated module over an Iwasawa algebra. This connection with Iwasawa theory, together with the proof of the Iwasawa main conjecture in this context by Mazur and Wiles, plays an important role in Huber and Kings' proof [33] of the Bloch-Kato conjecture for 1-dimensional representations of

Γ_{Q}

2. WHAT IS AN EULER SYSTEM?

For

K

a number field and

V

Γ_{K}

-representation as in Section 1, we have the notion of an Euler system for

V

, defined as follows. Let

S

be a finite set of places of

K

containing all infinite places, all primes above

p

and all primes at which

V

is ramified.

We define

R

to be the collection of integral ideals of

K

of the form

m = a \cdot b

, where a is a square-free product of primes of

K

not in

S

, and

b

divides

p^{\infty}

. For each

m \in R

, let

c [m]

be the ray class field modulo

m

. Then an Euler system for

(T, S)

is a collection of classes

c = {c [m] \in H^{1} (K [m], T) : m \in R}

satisfying the norm-compatibility relation

{cores}_{K [m]}^{K [q]} (c [m q]) = {\begin{cases} P_{q} (V^{*} (1), σ_{q}^{- 1}) \cdot c [m] & if q \notin S \\ c [m] & if q ∣ p \end{cases}

where cores denotes the Galois corestriction (or norm) map, and

σ_{q}

is the image of Frob

_{q}

Gal (K [m] / K)

. By an Euler system for

V

, we mean an Euler system for some

(T, S)

. (These general definitions are due to Kato, Perrin-Riou, and Rubin, building on earlier work of Kolyvagin; the standard reference is [56].)

The crucial application of Euler systems is the following: if an Euler system exists for

V

whose image in

H^{1} (K, V)

is non-zero (and

V

satisfies some auxiliary technical hypotheses), then we obtain a bound for the so-called relaxed Selmer group

^{1}

H_{r e l}^{1} (K, V) := \ker (H^{1} (K, V) \to \prod_{v ∤ p} H_{f}^{1} (K_{v}, V))

The relaxed Selmer group differs from the Bloch-Kato Selmer group in that we impose no local conditions at

p

. More generally, under additional assumptions on

V

and

c

, we can obtain finer statements taking into account local conditions at

p

, and hence control the dimension of the Bloch-Kato Selmer group itself.

Euler systems are hence extremely powerful tools for bounding Selmer groups, as long as we can understand whether the image of

c

H^{1} (K, V)

is non-vanishing. In order to

1 See [49] for this formulation. Theorem 2.2 .3 of [56] is an equivalent result, but expressed in terms of a Selmer group for

V^{*} (1)

, which is related to that of

V

by Poitou-Tate duality.
use an Euler system to prove new cases of the Bloch-Kato conjecture, one needs to establish a so-called explicit reciprocity law, which is a criterion for the non-vanishing of the Euler system in terms of the value

L (V^{*}, 1)

Challenges. In order to use Euler system theory to approach the Bloch-Kato conjecture, and other related problems such as the Iwasawa main conjecture, there are two major challenges to be overcome:

(1) Can we construct "natural" examples of Euler systems (satisfying appropriate local conditions), for interesting global Galois representations

V

(2) Can we prove reciprocity laws relating the images of these Euler systems in

H^{1} (K, V)

to the values of

L

-functions?

This was carried out by Kato [35] for the Galois representations arising from modular forms; but Kato's approach to proving explicit reciprocity laws has turned out to be difficult to generalise. More recently, in a series of works with various co-authors beginning with [40] (building on earlier work of Bertolini-Darmon-Rotger [6]), we developed a general strategy for overcoming these challenges, for Galois representations arising from automorphic forms for a range of reductive groups. We will describe this strategy in the remainder of this article.

Variants. A related concept is that of an anticyclotomic Euler system, in which

K

is a CM field, and we replace the ray-class fields

c [m]

with ring class fields associated to ideals of the real subfield

K^{+}

. These arise naturally when

V

is conjugate self-dual, i.e.

V^{c} = V^{*} (1)

where

c

denotes complex conjugation. The most familiar example is Kolyvagin's Euler system of Heegner points [39]; for more recent examples, see, e.g. [12, 15, 25]. Many of the techniques explained here for constructing and studying (full) Euler systems are also applicable to anticyclotomic Euler systems, and we shall discuss examples of both below.

A rather more distant cousin is the concept of a bipartite Euler system, which arises naturally in the context of level-raising congruences; cf. [31] for a general account, and [43] for a dramatic recent application to the Bloch-Kato conjecture. These require a rather different set of techniques, and we shall not discuss them further here.

The 1-critical condition. We conjectured in [49] that, in order to construct Euler systems for

V

by geometric means (i.e. as the images of motivic cohomology classes), we need to impose a condition on

V

: it needs to be 1 -critical.

However, our intended applications involve the Bloch-Kato conjecture for critical values of

L

-functions; so we need to construct Euler systems for representations that are 0 critical, rather than 1-critical. So we shall construct Euler systems for these representations in two stages: firstly, we shall construct Euler systems for auxiliary 1-critical representations

V

, using motivic cohomology; secondly, we shall "

p

-adically deform" our Euler systems, in order to pass from these 1 -critical

V

to others which are 0 -critical. This will be discussed in Section 4 below.

3. EULER SYSTEMS FOR SHIMURA VARIETIES

Shimura varieties. Let

(G, X)

be a Shimura datum, with reflex field

E

. For a level

K \subset

G (A_{f})

, we write

Y_{G} (K)

for the Shimura variety

{S h}_{K} (G, X)

. Our first goal will be to define Euler systems, either full or anticyclotomic, for Galois representations appearing in the Ã©tale cohomology of

Y_{G} (K)

. We shall attempt to give a systematic general treatment, but the reader should bear the following examples in mind:

(1)

G = {G L}_{2} \times_{{G L}_{1}} {G L}_{2}

, as in [4Ó©];

(2)

G = {Res}_{F / Q} {G L}_{2}

for

F

real quadratic, as in

[26, 41]

;

(3)

G = {G S p}_{4}

, as in [47];

(4)

G = {G S p}_{4} \times_{{G L}_{1}} {G L}_{2}

, as in [32];

(5)

G = G U (2, 1)

, as in

[48]

;

(6)

G = U (2 n - 1, 1)

for

n ⩾ 1

, as in [25].

Each of these groups is naturally equipped with a Shimura datum

(G, X)

. In examples (1)-(4), the reflex field

E

Q

; in (5) and (6), it is the imaginary quadratic field used to define the unitary group. (One can also retrospectively interpret Kato's construction [35] in these terms, taking

G = {G L}_{2}

; and similarly Kolyvagin's anticyclotomic Euler system [39], which is in effect the

n = 1

case of example (6).)

Ã‰tale cohomology. If

π

is a cuspidal automorphic representation which contributes to

H_{e t}^{*} (Y_{G} (K)_{\bar{E}}, V_{λ})

for some level

K

, where

V_{λ}

is the Ã©tale local system associated to the representation of

G

of highest weight

λ

, then we say

π

is cohomological in weight

λ

. It is conjectured that if this holds, then there exists a

p

-adic representation

ρ_{π}

Γ_{E}

, for each prime

p

and embedding

\bar{Q} ↪ {\bar{Q}}_{p}

, whose local Euler factors are determined by the Satake parameters of

π

at finite places, and whose Hodge-Tate weights are determined by

λ

For all of the above groups, the existence of such a

ρ_{π}

is known, and, moreover, if

π

is of "general type" (i.e. not a functorial lift from a smaller group), then the

π_{f}

-eigenspace in Ã©tale cohomology is concentrated in degree

d = \dim (X)

and isomorphic to

π_{f} \otimes ρ_{π}

. So we can find projection maps

H^{d} (Y_{G} (K)_{\bar{E}}, V_{λ}) \to ρ_{π}

, for a suitable choice of

K

. Via the Hochschild-Serre spectral sequence

H^{i} (E, H_{e t}^{j} (Y_{G} (K)_{\bar{E}}, V_{λ})) \Rightarrow H_{e t}^{i + j} (Y_{G} (K)_{E}, V_{λ})

we can thus obtain classes in the Galois cohomology of

ρ_{π}

as the images of classes in the

π_{f} -

eigenspace of

H_{e t}^{d + 1} (Y_{G} (K)_{E}, V_{λ})

. (For simplicity, we shall sketch the construction below assuming

λ = 0

, and refer to the original papers for the case of general coefficients.)

Motivic cohomology. In order to construct classes in

H_{e t}^{d + 1} (Y_{G} (K)_{E}, V_{λ})

, we shall use two other, related cohomology theories:

Motivic cohomology (see [3]), which takes values in $Q$ -vector spaces (or $Z$ -lattices in them), and is closely related to algebraic $K$ -theory and Chow groups;
Deligne-Beilinson (or absolute Hodge) cohomology (see [34]), which takes values in $R$ -vector spaces, and has a relatively straightforward presentation in terms of pairs $(ω, σ)$ , where $ω$ is an algebraic differential form, and $σ$ a real-analytic antiderivative of $Re (ω)$ .

There is no direct relation between Deligne-Beilinson cohomology and

p

-adic Ã©tale cohomology - one would not expect to compare vector spaces over

R

and over

Q_{p}

- but both of these cohomology theories have natural maps ("realisation maps") from motivic cohomology. So we shall use the following strategy, whose roots go back to [3]: we will write down elements of motivic cohomology whose images in Deligne-Belinson cohomology are related to values of

L

-functions; and we will consider the images of the same motivic cohomology classes in Ã©tale cohomology.

Pushforward maps. If

(H, y) ↪ (G, X)

is the inclusion of a sub-Shimura datum (with the same reflex field

E

), then we obtain finite morphisms of algebraic varieties over

E

Y_{H} (K \cap H)_{E} \to Y_{G} (K)_{E}

where

E

is the reflex field of

(H, y)

. More generally, for each

g \in G (A_{f})

we have a map

ι_{g} : Y_{H} {(g K g^{- 1} \cap H)}_{E} \to Y_{G} {(g K g^{- 1})}_{E} \overset{g}{\to} Y_{G} (K)

where the latter arrow is translation by

g

. So we have associated pushforward maps in all of our cohomology theories, namely

ι_{g, ⋆} : H_{m o t}^{j} (Y_{H} (K \cap H)_{E}, Z (t)) \to H_{m o t}^{j + 2 c} (Y_{G} (K)_{E}, Z (t + c))

for

j, r \in Z

, where

c = \dim X - \dim Y

(and similarly for Ã©tale cohomology with

Z_{p}

coefficients, or Deligne-Beilinson cohomology with

R

coefficients, compatibly with the realisation maps relating the theories).

We shall define motivic cohomology classes for

Y_{G} (K)

using the maps

ι_{g, ⋆}

. The compatibility of these classes with realisation functors allows us to compute the images of such classes in Deligne-Beilinson cohomology: the projection of such a class to the

π_{f} -

eigenspace will be computed using integrals over

Y_{H} (K \cap H) (C)

, involving the pullbacks of differential forms associated to cusp forms in the dual automorphic representation

π^{\lor}

Cycle classes and Siegel units. As an input to the above construction, we need a supply of "interesting" classes in

H_{mot}^{j} (Y_{H} (K \cap H), Z (r))

for some

k, r

which are in the image of motivic cohomology.

One possibility is to start with the identity class

1 \in H_{m o t}^{0} (Y_{H} (K \cap H), Z (0))

. The image of this class under

ι_{g, ⋆}

is the cycle class associated to the image of

ι_{g}

, a so-called "special cycle". This case is by no means trivial: indeed, these special cycles are the input used to define anticyclotomic Euler systems, such as Heegner points.

More subtly, one can obtain motivic cohomology classes from units in the coordinate ring of

Y_{H}

, using the relation

H_{m o t}^{1} (Y, Z (1)) = O (Y)^{\times}

for any variety

Y

. If

Y

is a modular curve (i.e. a Shimura variety for

{G L}_{2}

), then we have a canonical family of units: if

Y_{1} (N)

is the Shimura variety of level

{(\begin{array}{cc} * \\ 0 & 1 \\ 1 \end{array}) mod N}

, then we have the Siegel unit

z_{N} \in O {(Y_{1} (N))}^{\times}

denoted

_{c} g_{0, 1 / N}

in the notation of [35] (where

c

is an auxiliary integer coprime to the level). Crucially, we have an explicit formula for the image of this class in Deligne-Beilinson cohomology; it is given by

\begin{matrix} (3.1) & (d \log z_{N}, \log | z_{N} |) = (E_{2}, E_{0}^{a n} (0)) \end{matrix}

where

E_{2}

is a weight 2 Eisenstein series, and

E_{0}^{an} (s)

is a family of real-analytic Eisenstein series depending on a parameter

s \in C

. (See also [38] for analogues with coefficients, related to Eisenstein series of higher weights.)

Rankin-Eisenstein classes and Rankin-Selberg integrals. We shall consider the following general setting: we consider a Shimura datum

(H, y)

equipped with an embedding

ι : (H, y) \to (G, X)

, and also with a family of maps

ψ = (ψ_{1}, \dots, ψ_{t}) : (H, y) \to {({G L}_{2}, H)}^{t}

where

H

is the standard

{G L}_{2}

Shimura datum and

t ⩾ 1

. We then have a collection of classes

z_{N}^{H} = ψ_{1}^{*} (z_{N}) \cup \dots \cup ψ_{t}^{*} (z_{N}) \in H_{m o t}^{t} (Y_{H} (K_{H, 1} (N)), Z (t))

for some level

K_{H, 1} (N)

, which we call Eisenstein classes for

H

Remark 3.1. One might hope for a broader range of "Eisenstein classes" in motivic cohomology, associated to Eisenstein series on other groups which are not just copies of

{G L}_{2}

's. However, this question seems to be very difficult; see [21] for some results in this direction for symplectic groups. If we could construct motivic classes associated to Eisenstein series for the Siegel parabolic of

{G S p}_{2 n}

(rather than the Klingen parabolic as in [21]), or for the analogous parabolic subgroup in the unitary group

U (n, n)

, then it would open the way towards a far wider range of Euler system constructions.

By a motivic Rankin-Eisenstein class for

(G, X)

(with trivial coefficients), we shall mean a class of the form

ι_{g, ⋆} (_{c} z_{N}^{H}) \in H_{m o t}^{2 c + t} (Y_{G} (K)_{E}, Z (c + t))

for some

N

and some

g

and level

K

. If we choose our data

(H, y)

such that

2 c + t = 1 + d

, then these classes land in the cohomological degree we want. The twist

c + t

is then equal to

\frac{d + 1 + t}{2}

; hence, using the Hochschild-Serre spectral sequence, we can project the Ã©tale realisations of these classes into the groups

H^{1} (E, V_{π})

, where

V_{π} = ρ_{π} (\frac{d + 1 + t}{2})

Choosing the data

To define a Rankin-Eisenstein class, we need to choose the group

H

, and the maps

ι : H \to G

and

ψ : H \to {({G L}_{2})}^{t}

. To guide us in choosing these, we shall use "RankinSelberg-type" integral formulas for

L

-functions of automorphic representations. There are a wide range of such formulas, relating automorphic

L

-functions to integrals of the form

\int_{H (Q) Z_{G} (A) ∖ H (A)} ι^{*} (ϕ) ψ^{*} (E^{a n} (s_{1}) \times \dots \times E^{a n} (s_{n})) d h

where

E^{an} (s_{i})

are real-analytic

{G L}_{2}

Eisenstein series, and

ϕ

is a cuspform in the space of

π

. We call these period integrals. Typically, one expects such an integral to evaluate to a product of one or more copies of the

L

-function of

π

, evaluated at some linear combination of the parameters

s_{i}

. For instance, the Rankin-Selberg integral for

{G L}_{2} \times {G L}_{2}

is of this form, as is Novodvorsky's formula for the

L

-functions of

{G S p}_{4}

and

{G S p}_{4} \times {G L}_{2}

Using the explicit formula (3.1) relating Siegel units to Eisenstein series, one can often show that the Deligne-Beilinson realisations of Rankin-Eisenstein classes also lead to integrals of the form (3.2), for suitably chosen

ϕ

and

s_{i}

. When this applies, we can use it to relate our motivic Rankin-Eisenstein classes to special values of

L

-functions

^{2}

(as was carried out in Beilinson's original paper [3] for the

L

-functions of

{G L}_{2}

and

{G L}_{2} \times {G L}_{2}

; see, e.g.

[36, 42]

for more recent examples).

This gives one a guide to constructing "interesting" Rankin-Eisenstein classes for a given

(G, X)

: one first searches for a Rankin-Selberg integral describing the relevant

L

function, and then attempts to breathe motivic life into this real-analytic formula, interpreting it as the Deligne-Beilinson realisation of a motivic Rankin-Eisenstein class. One should hence interpret Rankin-Eisenstein classes as "motivic avatars" of Rankin-Selberg integral formulae.

In the anti-cyclotomic

(t = 0)

case, the period integral (3.2) is more mysterious; but there are still a number of results and conjectures predicting that these period integrals should be related to values of

L

-functions. For instance, the Gan-Gross-Prasad conjecture [22] gives such a relation in the important cases

S O (n) ↪ S O (n) \times S O (n + 1)

and

U (n) ↪

U (n) \times U (n + 1)

. This conjecture has recently been proved in the unitary case [9], although the orthogonal case is still open. We refer to Sakellaridis-Venkatesh [58] for conjectural generalisations to other pairs

(G, H)

Example 3.2. In our examples (1)-(6) above, we choose

H

and

t

as follows:

	$G$	$H$	$t$
(1)	${G L}_{2} \times {G L}_{2}$	${G L}_{2}$	1
(2)	${Res}_{F / Q} {G L}_{2}$	${G L}_{2}$	1
(3)	${G S p}_{4}$	${G L}_{2} \times_{{G L}_{1}} {G L}_{2}$	2
(4)	${G S p}_{4} \times {G L}_{2}$	${G L}_{2} \times_{{G L}_{1}} {G L}_{2}$	1
(5)	$G U (2, 1)$	${G L}_{2} \times_{{G L}_{1}} {Res}_{E / Q} {G L}_{1}$	1
(6)	$U (2 n - 1, 1)$	$U (n - 1, 1) \times U (n, 0)$	0

The integral formulae for

L

-functions underlying examples (1) and (2) are, respectively, the classical Rankin-Selberg integral and Asai's integral formula for quadratic Hilbert modular forms. Cases (3) and (4) are related to Novodvorsky's integral formula for

{G S p}_{4} \times {G L}_{2}

L

-functions (with an additional Eisenstein series on the

{G L}_{2}

factor in the former case); and case (5) to an integral studied by Gelbart and Piatetski-Shapiro in [23]. Example (6) is related to conjectures of Getz-Wambach [24] on Friedberg-Jacquet periods for automorphic representations of unitary groups.

Rankin-Eisenstein classes and norm relations. In order to build Euler systems (either full or anticyclotomic) from Rankin-Eisenstein classes, we need the following conditions to hold:

("Open orbit" condition) The group $H$ has an open orbit on the product

(G / B_{G}) \times {(P^{1})}^{t}

where

B_{G}

is a Borel subgroup of

G

, and

H

acts on

G / B_{G}

via

ι

, and on

{(P^{1})}^{t}

via

ψ

("Small stabiliser" condition) For a point $u$ in the open orbit, let $S_{u}$ be the subgroup of $H$ which fixes $u$ and acts trivially on the fibre at $u$ of the tautological ${(G_{m})}^{t}$ -bundle over ${(P^{1})}^{t}$ . Then we require that the image of $S_{u}$ has small image in the maximal torus quotient of $H$ .

The role of the "small stabiliser" condition is to allow us to construct classes over field extensions. Since the connected components of the Shimura variety

Y_{G}

are defined over abelian extensions of

E

, and the Galois action on the component group is described by class field theory, we can modify the Rankin-Eisenstein classes to define elements in

H_{e t}^{d + 1} (Y_{G} (K)_{F}, Z_{p} (\frac{d + 1 + t}{2}))

for a fixed level

K

and varying abelian extensions

F / E

. The class of abelian extensions that arise will depend on the image of

S_{u}

in the maximal torus quotient; in the examples (1)-(5) above, since

S_{u} = {1}

and the splitting field of the Galois action is the full maximal abelian extension of

E

, so we obtain classes over all ray class fields of

E

. On the other hand, in example (6) we obtain only the anticyclotomic extension (as one would expect, since

t = 0

in this case).

The "open orbit" condition allows us to prove a so-called vertical norm relation, showing that after applying Hida's ordinary projector, the Rankin-Eisenstein classes form norm-compatible families over the tower

E [p^{\infty}] / E

, with uniformly bounded denominators relative to the Ã©tale cohomology with

Z_{p}

-coefficients. This machinery is worked out in considerable generality in [44,46]; the arguments also simultaneously show that the RankinEisenstein classes interpolate in Hida-type

p

-adic families (in which the weight

λ

π

varies).

A much more subtle problem is that of horizontal norm relations, comparing classes over

E [m]

and

E [m q]

for auxiliary primes

q ∤ m

, with the Euler factors

P_{q}

appearing in the comparison. The strategy developed in [47] and refined in [48] is to use multiplicity-one results in smooth representation theory to reduce the norm relation to a purely local calculation with zeta-integrals, which can then be computed explicitly to give the Euler factor. These multiplicity-one results are themselves closely bound up with the open-orbit condition; see

[57

Remark 3.3. The open-orbit condition, together with the assumption that

2 c + t = 1 + d

, amount to stating that the diagonal map

(ι, ψ) : (H, y) ↪ (G, X) \times {({G L}_{2}, H)}^{t}

is a special pair of Shimura data in the sense of [59, DEFINITION 3.1]. We can thus interpret the "small stabiliser" condition, at least for

t = 0

, as a criterion for when the special cycles studied in [59] extend to norm-compatible families over field extensions.

4. DEFORMATION TO CRITICAL VALUES

Critical values. The above methods allow us to define Euler systems for the automorphic

Galois representations

V_{π} = ρ_{π} (\frac{d + 1 + t}{2})

, where

π

is cohomological in weight 0 ; and there are generalisations to representations which are cohomological for a certain range of nonzero weights

λ

, determined by branching laws for the restriction of algebraic representations from

\tilde{G} = G \times {({G L}_{2})}^{t}

H

. Let us write

Σ_{1}

for the set of weights

λ

which are accessible by these methods, for some specific choice of

H

and

ψ

; this is a convex polyhedron in the weight lattice of

G

, cut out by finitely many linear inequalities. In the examples (1)-(5), one checks that for any

π

whose weight lies in

Σ_{1}

, the representation

V_{π}

is 1-critical, consistently with the conjectures of [49].

However, our goal is to prove the Bloch-Kato conjecture for critical

L

-values; so we are interested in those

λ

such that, for

π

of weight

λ

, the representation

V

is 0 -critical, so

L (π^{\lor}, \frac{1 - t}{2})

is a critical value. The set of such

λ

is a finite disjoint union of polyhedral regions; and we let

Σ_{0}

be one of these regions, chosen to be adjacent to

Σ_{1}

. In order to approach the Bloch-Kato conjecture, we need to find a way of "deforming" our Euler systems from

Σ_{1}

Σ_{0}

Example 4.1. For

G = {G L}_{2} \times {G L}_{2}

, the Galois representations associated to cohomological representations of

G

have the form

(ρ_{f} \otimes ρ_{g}) (n)

, where

f, g

are modular forms (of some weights

k + 2, ℓ + 2

with

k, ℓ ⩾ 0

) and

n

is an arbitrary integer. If we set

j = k + ℓ + 1 - n

,
then the set

Σ_{1}

is given by the inequalities

{0 ⩽ j ⩽ min (k, ℓ)}

and there are two candidates for the set

Σ_{0}

, namely

{ℓ + 1 ⩽ j ⩽ k} and {k + 1 ⩽ j ⩽ ℓ}

Remark 4.2. A slightly different, but related, numerology applies for anticyclotomic Euler systems. In these cases, the relevant

L

-value is always critical, but it lies at the centre of the functional equation, so it may be forced to vanish for sign reasons. Since the local root numbers at the infinite places depend on

λ

, we have some ranges of weights where the root number is +1 (where we expect interesting central

L

-values) and others where it is -1 (where we expect anticyclotomic Euler systems). These play the roles of the 0 -critical and 1 -critical regions in the case of full Euler systems.

The Bertolini-Darmon-Prasanna strategy. Although the " 0 -critical" and " 1 -critical" weight ranges are disjoint, we can relate them together

p

-adically, using a strategy introduced by Bertolini, Darmon and Prasanna in [5].

The weights

λ

of cohomological representations can naturally be seen as points of a

p

-adic analytic space

W

(parametrising characters

T (Z_{p}) \to C_{p}^{\times}

, where

T

is a maximal torus in

G

). This space is isomorphic to a finite union of

n

-dimensional open discs, where

n

is the rank of

G

. Crucially, both

Σ_{0}

and

Σ_{1}

are Zariski-dense in

W

Hida theory shows that there is a finite flat covering

E \to W

, the ordinary eigenvariety of

G

, whose points above a dominant integral weight

λ

("classical points") biject with automorphic representations

π

G

which are cohomological of weight

λ

and

p

-ordinary.

We thus have two separate families of objects, indexed by different sets of classical points on

E

at points whose weights lie in $Σ_{0}$ , we have the critical values of the complex $L$ -function;
at points whose weights lie in $Σ_{1}$ , we have Euler systems arising from motivic cohomology.

Our first goal will be to "analytically continue" the Euler system classes from

Σ_{1}

into

Σ_{0}

This is not all that we require, however, since we also need a relation between the resulting Euler system for each 0 -critical

V

and the

L

-value

L (V^{*}, 1)

. Relations of this kind are known as explicit reciprocity laws, and they are the crown jewels of Euler system theory. Following a strategy initiated in [5] and further developed in [37], in order to prove explicit reciprocity laws, we shall use a second kind of

p

-adic deformation: besides deforming Euler system classes from

Σ_{1}

Σ_{0}

, we shall also deform

L

-values from

Σ_{0}

into

Σ_{1}

. The strategy consists of the following steps:

(i) We shall construct a function on the eigenvariety - an "analytic

p

-adic

L

function" - whose values in

Σ_{0}

are critical

L

-values (modified by appropriate periods and Euler factors).
(ii) Using the Perrin-Riou regulator map of

p

-adic Hodge theory, we construct a second analytic function on the eigenvariety - a "motivic

p

-adic

L

-function" - whose value at some cohomological

π

measures the non-triviality of Euler system classes for

π

locally at

p

Note that the motivic

p

-adic

L

-function has no a priori reason to be related to complex

L

-values; however, its values in

Σ_{1}

are by definition related to the Euler system classes (which arise from motivic cohomology, hence the terminology).

(iii) We shall prove a "

p

-adic regulator formula", showing that the values of the analytic

p

-adic

L

-function in at points in

Σ_{1}

are related to the localisations of the Euler system classes at

p

(iv) Using the regulator formula of step (iii), we can deduce that the motivic and analytic

p

-adic

L

-function coincide at points in

Σ_{1}

. Since weights lying in

Σ_{1}

are Zariski-dense in

E

, this implies the two

p

-adic

L

-functions coincide in

Σ_{0}

as well. Since the values of the analytic

p

-adic

L

-function in

Σ_{0}

are complex

L

-values, we obtain the sought-for explicit reciprocity law.

At the time of writing, this strategy has only been fully carried out for the examples (1) and (3) in our list, and partially for (4). However, the remaining cases are being treated in ongoing work of members of our research groups; and we expect the strategy to extend to many other Euler systems (both full and anticyclotomic) besides these.

5. CONSTRUCTING $p$ -ADIC $L$ -FUNCTIONS

Coherent cohomology

To construct the analytic

p

-adic

L

-function, we shall use the integral formula (3.2). Previously, for weights in

Σ_{1}

, we interpreted this integral as a cup-product in DeligneBeilinson cohomology. We shall now give a different cohomological interpretation of the same formula, for weights in the range

Σ_{0}

. Following a strategy introduced by Harris [28,29], we can choose the cusp-form

ϕ

, and the Eisenstein series, to be harmonic differential forms (with controlled growth at the boundary) representing Dolbeault cohomology classes valued in automorphic vector bundles. These can then be interpreted algebraically, via the comparison between Dolbeault cohomology and Zariski sheaf cohomology. The upshot is that

L (π^{\lor}, \frac{1 - t}{2})

can be related to a cup-product in the cohomology of coherent sheaves on a smooth toroidal compactification

{S h}_{K} (H, y)_{Σ}^{tor}

{S h}_{K} (H, y)

Interpolation

In order to construct a

p

-adic

L

-function, we need to show that the cohomology classes appearing in our formula for the

L

-function interpolate in Hida-type

p

-adic families, and that the cup-product of these families makes sense.

Until recently, there was a fundamental limitation in the available techniques: we could only interpolate cohomology classes corresponding to holomorphic automorphic forms (i.e. degree 0 coherent cohomology), or (via Serre duality) those in the top-degree cohomology, which correspond to anti-holomorphic forms. This is an obstacle for our intended applications, since the integral formulas relevant for Euler systems always involve coherent cohomology in degrees close to the middle of the possible range. (More precisely, the relevant degree is

\frac{d + t - 1}{2}

, where

t

is the number of Eisenstein series present, which is typically

0, 1, 2

.) So unless

d

is rather small, using holomorphic or anti-holomorphic classes will not work.

A slightly wider range of "product type" examples arises when

(G, X)

is a product of two Shimura data

(G_{1}, X_{1}) \times (G_{2}, X_{2})

of approximately equal dimension, with

\dim (X_{1}) - \dim (X_{2}) = t - 1

; then we can build a class in the correct degree as the product of an anti-holomorphic form on

X_{1}

and a holomorphic one on

X_{2}

, and the resulting cup-products can often be understood as Petersson-type scalar products in Hida theory. For instance, the Rankin-Selberg integral formula can be analysed in this way [30]. However, for

G = {G S p}_{4}

(with

t = 2

and

d = 3

), we need to work with a class in coherent

H^{2}

, and these are not seen by orthodox Hida theory.

Higher Hida theory. A beautiful solution to this problem is provided by the "higher Hida theory" developed in [55]. Pilloni's work shows that degree 1 coherent cohomology for the

{G S p}_{4}

Shimura variety interpolates in a "partial" Hida family, with one weight fixed and the other varying

p

-adically.

A key ingredient in this work is to consider a certain stratification of the

mod p

fibre of the

{G S p}_{4}

Shimura variety

Y_{G}

(for some level structure unramified at

p

). This space parametrises abelian surfaces

A

with a principal polarisation and some prime-to-

p

level structure. There is an open subspace

Y_{G}^{ord}

, whose complement has codimension 1, where

A

is ordinary; and a slightly larger open set, with complement of codimension 2 , where the

p

-rank of

A

(the dimension of the multiplicative part of

A [p]

) is

⩾ 1

. This stratification can be extended to a toroidal compactification

X_{G}

Y_{G}

; and Pilloni's approach to studying

H^{1}

p

-adic families is based on restricting to the tube of this

p

-rank

⩾ 1

locus in the

p

adic completion

X_{G}

X_{G}

. (In contrast, orthodox Hida theory for

{G S p}_{4}

involves working over the ordinary locus; this is very effective for studying

H^{0}

but disastrous for studying

H^{1}

, since the ordinary locus is affine in the minimal compactification, so its cuspidal cohomology vanishes in positive degrees.)

In [45] we carried out a (slightly delicate) comparison of stratifications, showing that we can find an embedding

ι_{g}

of an

H

-Shimura variety, for a carefully chosen

g

, so that the preimage of the

p

-rank

⩾ 1

locus in

X_{G}

is the ordinary locus in

X_{H}

, that is, the image of

X_{H}

"avoids" the locus where the

p

-rank is exactly 1 . Using this, we constructed pushforward maps from the orthodox

(H^{0})

Hida theory for

H

to Pilloni's

H^{1}

theory for

G

, interpolating the usual coherent-cohomology pushforward maps for varying weights. This is the tool we need to construct analytic

p

-adic

L

-functions for

{G S p}_{4}

and for

{G S p}_{4} \times {G L}_{2}

At present higher Hida theory, in the above sense, is only available for a few specific groups, although these include many of the ones relevant for Euler systems: besides

{G S p}_{4}

, the group

G U (2, 1)

is treated in [53], and Hilbert modular groups in [27] (in both cases assuming

G

is locally split at

p)

. In the

{G S p}_{4}

and

G U (2, 1)

cases the results are also slightly weaker than one might ideally hope, since we only obtain families in which one component of the weight is fixed and the others vary (so the resulting

p

-adic

L

-functions have one variable fewer than one would expect). However, we expect that these restrictions will be lifted in future work.

Remark 5.1. A related theory, higher Coleman theory, has been developed by Boxer and Pilloni in [13]. This theory also serves to interpolate higher-degree cohomology in families, with all components of the weight varying; and the theory applies to any Shimura variety of abelian type. However, unlike the higher Hida theory of [55], this theory only applies to cohomology classes satisfying an "overconvergence" condition. This rules out the 2-parameter

{G L}_{2}

Eisenstein family which plays a prominent role in the constructions of [45], as this Eisenstein series is not overconvergent. It may be possible to work around this problem by combining the higher Coleman theory of [13] with the theory of families of nearly-overconvergent modular forms for

{G L}_{2}

introduced by Andreatta-Iovita [1]; but the technical obstacles in carrying this out would be formidable.

6. P-ADIC REGULATORS

We now turn to step (iii) of the BDP strategy: relating values of the analytic

p

-adic

L

-function in the range

Σ_{1}

to the localisations at

p

of the Euler system classes.

Syntomic cohomology. For all but finitely many primes, the Shimura variety has a smooth integral model over

Z_{p}

, and the motivic Rankin-Eisenstein classes can be lifted to the cohomology of this integral model. This allows us to study them via another cohomology theory, Besser's rigid syntomic cohomology [7]. This is a cohomology theory for smooth

Z_{p}

-schemes

y

, which has two vital properties:

Via works of Fontaine-Messing and NizioÅ‚, one can define a comparison map relating syntomic cohomology of $y$ to Ã©tale cohomology of its generic fibre $Y$ ; and this is compatible with motivic cohomology, in the sense that we have a commutative diagram (see [8]):

where the map

r_{ett}

is the Ã©tale realisation map.

The Fontaine-Messing-Niziol map induces the Bloch-Kato exponential map on Galois cohomology; so, for a class in

H_{e t}^{*} (Y, n)

in the image of motivic cohomol-
ogy of

y

, one can express its Bloch-Kato logarithm via cup-products in syntomic cohomology ("syntomic regulators").

Rigid syntomic cohomology and its variant, fp-cohomology, were defined by Besser as a generalisation of Coleman's theory of $p$ -adic integration. It is computed by an explicit complex of sheaves which is a $p$ -adic analogue of the realanalytic Deligne-Beilinson complex: sections of this complex are pairs $(ω, σ)$ , where $ω$ is an algebraic differential form, and $σ$ is an overconvergent rigidanalytic differential form such that $d σ = (1 - φ) ω$ , where $φ$ is a local lift of the Frobenius of the special fibre.

In a series of works, beginning with the breakthrough [16] by Darmon-Rotger (see also

[6, 10, 38]

), rigid syntomic cohomology has been systematically exploited to compute the Bloch-Kato logarithms of Rankin-Eisenstein classes when

G

is a product of copies of

{G L}_{2}

, in terms of Petersson products of (non-classical)

p

-adic modular forms. These can then be interpreted as values of

p

-adic

L

-functions in a "1-critical" region

Σ_{1}

. All of these

p

-adic

L

-functions are "product type" settings in the sense explained above, involving coherent cohomology in either top or bottom degree.

Remark 6.1. A key role in these constructions is played by an explicit formula for the image of the Siegel unit in the syntomic cohomology of the ordinary locus of the modular curve, which is the

p

-adic counterpart of equation (3.1): it is represented by the pair

(dlog z_{N}, (1 - φ) \log_{p} z_{N}) = (E_{2}, E_{0}^{(p)})

where

E_{0}^{(p)}

is a

p

-adic Eisenstein series of weight 0 .

We can thus understand these syntomic regulator formulae as

p

-adic counterparts of the integral formula (3.2), with the integral understood via Coleman's

p

-adic integration theory, and the real-analytic Eisenstein class replaced by a

p

-adic one.

The

{G S p}_{4}

regulator formula. The approach to computing regulators of Ã©tale classes via syntomic cohomology generalises to Euler systems for other Shimura varieties, such as

{G S p}_{4}

: one can always express the image of the Euler system class under the Bloch-Kato logarithm, paired against a suitable de Rham cohomology class (lying in the

π^{\lor}

-eigenspace), as a cup product in syntomic cohomology.

However, syntomic cohomology of the whole Shimura variety is not well-suited to explicit computations, since there is generally no global lift of the Frobenius of the special fibre. The first major problem is hence to express the pairing in terms of the syntomic cohomology of certain open subschemes of the Shimura variety which do possess an explicit Frobenius lift. This requires some results on the Hecke eigenspaces appearing in the rigid cohomology of Newton strata of the special fibre, which are the

ℓ = p

counterparts of the vanishing theorems proved by Caraiani-Scholze [14] for

ℓ

-adic cohomology for

ℓ \neq p

The second major problem is to establish a link between rigid syntomic cohomology and coherent cohomology, for varieties admitting a Frobenius lifting. We succeeded in proving such a relation via a new a spectral sequence (the so-called PoznaÅ„ spectral sequence)
which is a syntomic analogue of the Hodge-de Rham spectral sequence: its

E_{1}

page is given by the mapping fibre of

1 - φ

on coherent cohomology, and its abutment is rigid syntomic cohomology. In the case of the ordinary locus of the modular curve, where all coherent cohomology in positive degrees vanishes, this reduces to the description of a syntomic class as a pair of global sections

(ω, σ)

as described above.

Thanks to this new spectral sequence, we were able to express the syntomic regulator of our Rankin-Eisenstein class for

{G S p}_{4}

as a pairing in coherent cohomology, which we could identify as a specialisation of the pairing in higher Hida theory defining the

p

-adic

L

-function. We can hence identify the logarithm of the

{G S p}_{4}

Euler system class with a noncritical value of a

p

-adic

L

-function. This is the first example of a

p

-adic regulator formula where the

p

-adic

L

-function is not of product type. We expect this strategy to be applicable to all the other Euler systems mentioned in Section 3 above. Cases (2) and (6) are currently work in progress by Giada Grossi, and by Andrew Graham and Waqar Shah, respectively; and case (5) is being explored by some members of our research groups.

7. DEFORMATION TO CRITICAL VALUES

We can now proceed to the final step of the Bertolini-Darmon-Prasanna strategy: deforming from

Σ_{1}

Σ_{0}

First, we must show that Euler systems interpolate over the eigenvariety. The Ã©tale cohomology eigenspaces attached to cohomological,

p

-ordinary automorphic representations are known to interpolate in families, giving rise to sheaves of Galois representations over

E

. With this in hand, the machinery of

[44, 46]

then shows that the Euler system classes themselves interpolate, giving families of Euler systems taking values in these sheaves.

A generalisation of Coleman and Perrin-Riou's theory of "big logarithm" maps (cf. [37]) also allows us to define a motivic

p

-adic

L

-function

L^{mot}

associated to the bottom class in our family of Euler systems. Perrin-Riou's local reciprocity formula implies that

L^{mot}

has an interpolation property both in

Σ_{0}

and in

Σ_{1}

. For classical points

π

whose weights lie in

Σ_{1}

, the value of

L^{mot}

interpolates the Bloch-Kato logarithm of the geometrically-defined Euler system class for

V_{π}

. Much more subtly, if we evaluate

L^{mot}

at points

π

whose weights lie in

Σ_{0}

, it computes the image under the dual-exponential map of the bottom class in the Euler system for

V_{π}

which we have just defined using analytic continuation.

We would like to make the following argument: "the regulator formula shows that

L^{mot}

and the analytic

p

-adic

L

-function

L

agree at points in

Σ_{1}

, and these are Zariskidense; so

L = L^{mot}

everywhere". This is essentially how we proved an explicit reciprocity law for

{G L}_{2} \times {G L}_{2}

in [37]. Unfortunately, there are two subtle technical hitches which occur in making this argument precise for

{G S p}_{4}

The first is that

L

and

L^{mot}

take values in different line bundles over the eigenvariety

E

(one interpolating coherent cohomology, and the other

D_{cris}

of a certain subquotient of Ã©tale cohomology). At each classical point of

E

, we have a canonical isomorphism between the fibres of these two line bundles; but it is far from obvious a priori that these "pointwise" isomorphisms at classical points interpolate into an isomorphism of line bundles. For the

{G L}_{2}

ordinary eigenvariety, we do have such an isomorphism, the

p

-adic Eichler-Shimura isomorphism of Ohta [54] (extended to non-ordinary families in [2]). However, the case of higher-dimensional Shimura varieties such as

{G S p}_{4}

is more difficult: one expects several Eichler-Shimura isomorphisms, each capturing coherent cohomology in a different degree, and at present only the case of

H^{0}

is available in the literature [19]. For the problem at hand, it is the coherent

H^{1}

(and dually

H^{2}

) which is relevant.

The second is that, while

Σ_{1}

is indeed Zariski-dense in the eigenvariety, the function

L

is only defined on a lower-dimensional "slice" of the eigenvariety (on which the

{G S p}_{4}

form has weight

(r_{1}, r_{2})

, with

r_{1}

varying and

r_{2}

fixed), and the intersection of each individual slice with

Σ_{1}

is not Zariski-dense in the slice.

In [50], we circumvented these problems in a somewhat indirect way, by appealing to a second, independent construction of an analytic

p

-adic

L

-function, defined using Shalika models for

{G L}_{4}

[20]. As written this construction shares with [45] the shortcoming of requiring

r_{2}

to be fixed, but the methods of [44] can be applied in order to extend this construction by varying

r_{2}

as well. Using this we were able to

The lack of an Eichler-Shimura isomorphism in families - or, more precisely, of an isomorphism between the sheaves in which

L^{m o t}

and the

{G L}_{4} p

-adic

L

-function take values - can be dealt with via the so-called "leading term argument". This proceeds as follows. There is clearly a meromorphic isomorphism between these sheaves which maps one

p

-adic

L

-function to the other (since both are clearly non-zero).

^{3}

If this meromorphic isomorphism degenerates to zero at some "bad" 0 -critical

π

, then the bottom class in our Euler system for

π

lies in the kernel of the Perrin-Riou regulator. However, this would also apply to all the classes

c [m]

in this Euler system, for all values of

m

. So we obtain an Euler system satisfying a very strong local condition at

p

; and a result of Mazur-Rubin [52] shows that this condition is so strong that it forces the entire Euler system to be zero. Hence we can replace all of these classes by their derivatives in the weight direction, which amounts to renormalising the Eichler-Shimura map to reduce its order of vanishing by 1 .

Iterating this process, we eventually obtain a non-trivial Euler system for

π

; and if

L (π^{\lor}, \frac{1 - t}{2}) \neq 0

, the bottom class of this Euler system is non-zero. We can now deduce the vanishing of

H_{f}^{1} (Q, V_{π})

, where

V_{π} = ρ_{π} (\frac{d + 1 + t}{2})

, as predicted by the Bloch-Kato conjecture.

Non-regular weights

The above strategy can also be used to study automorphic forms which are not cohomological (so

π

does not contribute to Ã©tale cohomology), as long as

π

contributes to coherent cohomology in the correct degree. For instance, this applies to weight 1 modular forms, which is crucial in several works such as [17] which use Euler systems to study the Birch-Swinnerton-Dyer conjecture for Artin twists of elliptic curves. It also applies to paramodular Siegel modular forms for

{G S p}_{4}

of parallel weight 2 , such as those corresponding to paramodular abelian surfaces.

In this situation, if

π

is ordinary at

p

, it follows from the results of [13] that it defines a point on the eigenvariety

E

. However, in contrast to the case of cohomological weights, it is not clear if the eigenvariety is smooth, or Ã©tale over weight space, at

π

; results of BellaicheDimitrov show that this can fail even for

{G L}_{2}

[4].

A

is a paramodular abelian surface over

Q

which is ordinary at

p

, and has analytic rank 0 , then we can use the above approach to prove the finiteness of

A (Q)

(as predicted by the Birch-Swinnerton-Dyer conjecture), and of the

p

-part of the Tate-Shafarevich group, under the assumption that the

{G S p}_{4}

eigenvariety be smooth at the point corresponding to

A

. This is work in progress.

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DAVID LOEFFLER

Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK, d.a.loeffler@warwick.ac.uk

SARAH LIVIA ZERBES

Department of Mathematics, University College London, London WC1E 6BT, UK, and ETH ZÃ¼rich, Switzerland, s.zerbes@ucl.ac.uk

COUNTING PROBLEMS: CLASS GROUPS, PRIMES, AND NUMBER FIELDS

LILLIAN B. PIERCE

ABSTRACT

Each number field has an associated finite abelian group, the class group, that records certain properties of arithmetic within the ring of integers of the field. The class group is well studied, yet also still mysterious. A central conjecture of Brumer and Silverman states that for each prime

ℓ

, every number field has the property that its class group has very few elements of order

ℓ

, where "very few" is measured relative to the absolute discriminant of the field. This paper surveys recent progress toward this conjecture, and outlines its close connections to counting prime numbers, counting number fields of fixed discriminant, and counting number fields of bounded discriminant.

MATHEMATICS SUBJECT CLASSIFICATION 2020

Primary 11R29; Secondary 11R45, 11N05

KEYWORDS

Class groups, counting number fields, distribution of primes

1. HISTORICAL PRELUDE

In a 1640 letter to Mersenne, Fermat stated that an odd prime

p

satisfies

p = x^{2} + y^{2}

if and only if

p \equiv 1 (mod 4)

. Roughly 90 years later, Euler learned of Fermat's statement through correspondence with Goldbach, and by 1749, he worked out a proof. This fits into a bigger question, which Euler studied as well: for each

n \geq 1

, which primes can be written as

p = x^{2} + n y^{2}

? Even more generally: which binary quadratic forms

a x^{2} + b x y + c y^{2}

represent a given integer

m

? This question also motivated work of Lagrange and Legendre, and then appeared in Gauss's celebrated 1801 work Disquisitiones Arithmeticae; see [26].

Gauss partitioned binary quadratic forms of discriminant

D = b^{2} - 4 a c

into equivalence classes under

{S L}_{2} (Z)

changes of variable. (Here we will speak only of fundamental discriminants

D

; for notes on the original setting, see [84].) Gauss showed that for each

D

there are finitely many such classes (the cardinality is the class number, denoted

h (D)

), and verified that the classes obey a group law (composition). Based on extensive computation, Gauss noticed that as

D \to - \infty

, small class numbers stopped appearing, writing: "Nullum dubium esse videtur, quin series adscriptae revera abruptae sint...Demonstrationes autem rigorosae harum observationum perdifficiles esse videntur." ("It seems beyond doubt that the sequences written down do indeed break off... However, rigorous proofs of these observations appear to be most difficult" [43, P. 13].) As

D \to + \infty

, a quite different behavior seemed to appear, leading to a conjecture that

h (D) = 1

for infinitely many

D > 0

It is hard to exaggerate the interest these two conjectures have generated. In the 1830s, Dirichlet proved a class number formula, relating the class number

h (D)

of a (fundamental) discriminant

D

to the value

L (1, χ)

of an

L

-function associated to a real primitive character

χ

modulo

D

. Consequently, throughout the 1900s, Gauss's questions were studied via the theory of the complex-variable functions

L (s, χ)

. A remarkable series of works by Hecke, Deuring, Mordell, and Heilbronn confirmed that for

D < 0

the class number

h (D)

attains any value only finitely many times. How many times? Famously, the work of Heegner, Baker, and Stark proved that there are 9 (fundamental) discriminants

D < 0

with class number 1. In full generality, Goldfeld showed an effective lower bound for

h (D)

when

D < 0

would follow from a specific case of the Birch-Swinnerton-Dyer conjecture, which was then verified by Gross and Zagier; see [42]. Now, for each

1 \leq N \leq 100

, one may find the number of discriminants

D < 0

with

h (D) = N

in [93]. As for the other conjecture, that infinitely many (fundamental) discriminants

D > 0

have class number 1 , this remains open, and very mysterious. These historical antecedents hint at the intertwined currents of "counting" and the analytic study of

L

-functions, which will also be present in the work we will survey.

We briefly mention another historical motivation for the study of class numbers, namely the failure of unique factorization. For example, in the ring

Z [\sqrt{- 5}], 21 = 3 \cdot 7

but it also factors into irreducible, nonassociated factors as

(1 + 2 \sqrt{- 5}) (1 - 2 \sqrt{- 5})

. Here is a problem where the failure of unique factorization has an impact. Suppose one is searching for solutions

x, y, z \in N

to the equation

x^{p} + y^{p} = z^{p}

for a prime

p \geq 3

. If a nontrivial
solution

(x, y, z)

exists, then for

ζ_{p}

p

th root of unity, we could write

y \cdot y \dots y = (z - x) (z - ζ_{p} x) \dots (z - ζ_{p}^{p - 1} x)

Z [ζ_{p}]

possesses unique factorization, two such factorizations cannot exist, so

(x, y, z)

cannot exist-verifying Fermat's Last Theorem for this exponent

p

. But to the disappointment of many, unique factorization fails in

Z [ζ_{p}]

for infinitely many

p

. As Neukirch writes, "Realizing the failure of unique factorization in general has led to one of the grand events in the history of number theory, the discovery of ideal theory by Eduard Kummer" [69, cH. I Â§3].

1.1. The class group

Let

K / Q

be a number field of degree

n

, with associated ring of integers

O_{K}

. Every proper integral ideal

a \subset O_{K}

factors into a product of prime ideals

p_{1} \dots p_{k}

in a unique way (salvaging the notion of unique factorization). Moreover, the fractional ideals of

K

form an abelian group

J_{K}

, the free abelian group on the set of nonzero prime ideals of

O_{K}

. In the case that every ideal in

J_{K}

belongs to the subgroup

P_{K}

of principal ideals,

O_{K}

is a principal ideal domain, and unique factorization holds in

O_{K}

. But more typically, some "expansion" occurs when passing to ideals; the class group of

K

is defined to measure this.

The class group of

K

is the quotient group

{C l}_{K} = J_{K} / P_{K}

The elements in

{C l}_{K}

are ideal classes, and the cardinality

| {C l}_{K} |

is the class number. The quotient

J_{K} / P_{K}

is trivial (so that every ideal is a principal ideal, and

| {C l}_{K} | = 1

) precisely when unique factorization holds in

O_{K}

. (Thus the above strategy for Fermat's Last Theorem works for

p

| {C l}_{Q (ζ_{p})} | = 1

. In fact, Kummer showed that as long as the class number of

Q (ζ_{p})

is indivisible by

p

, the argument can be salvaged; see [31]. Such a prime is called a regular prime. Here is an open question: are there infinitely many regular prime numbers?)

By a result of Minkowski in the geometry of numbers, every ideal class in

{C l}_{K}

contains an integral ideal

b

with norm

ℜ (b) = (O_{K} : b)

satisfying

\begin{matrix} (1.1) & ℜ (b) \leq (2 / π)^{s} \sqrt{D_{K}} \end{matrix}

where

D_{K} = | Disc (K / Q) |

and

s

counts the pairs of complex embeddings of

K

. As there are finitely many integral ideals of any given norm, Landau deduced (see [68, Ñ‚Ð½M. 4.4]):

\begin{matrix} (1.2) & | {C l}_{K} | ≪_{n} D_{K}^{1 / 2} \log^{n - 1} D_{K} \end{matrix}

In particular, the class group of a number field

K

is always a finite abelian group. (Throughout,

A ≪_{κ} B

indicates that there exists a constant

C_{κ}

such that

| A | \leq C_{κ} B

When

K = Q (\sqrt{D})

is a quadratic field, this relates in a precise way to Gauss's construction of the class number for binary quadratic forms of discriminant

D

(see [8]). In modern terms, Gauss asked whether for each

h \in N

, there are finitely many imaginary quadratic fields

K

with

| {C l}_{K} | = h

? (Yes.) Are there infinitely many real quadratic fields

K

with

| {C l}_{K} | = 1

? (We do not know.) In fact, here is an open question: are there infinitely many number fields, of arbitrary degrees, with class number 1? Here is another open question: are
there infinitely many number fields, of arbitrary degrees, with bounded class number? These difficult questions must consider the regulator

R_{K}

of the field

K

, due to the (ineffective) inequalities by Siegel (for quadratic fields) and Brauer (in general) [68, ÑÐ½. 8]:

D_{K}^{1 / 2 - ε} ≪_{n, ε} | {C l}_{K} | R_{K} ≪_{n, ε} D_{K}^{1 / 2 + ε}, for all ε > 0

2. THE $ℓ$ -TORSION CONJECTURE

In addition to studying the size of the class group, it is also natural to study its structure. We will focus on the

ℓ

-torsion subgroup, defined for each integer

ℓ \geq 2

{C l}_{K} [ℓ] = {[a] \in {C l}_{K} : [a]^{ℓ} = I d}

For example, the class number is divisible by a prime

ℓ

precisely when

| {C l}_{K} [ℓ] | > 1

. Related problems include studying the exponent of the class group, or counting how many number fields of a certain degree have class number divisible, or indivisible, by a given prime

ℓ

. Such problems are addressed for imaginary quadratic fields in

[4, 44, 45, 82]

In this survey, we will focus on upper bounds for the

ℓ

-torsion subgroup. The Minkowski bound (1.2) provides an upper bound for any field of degree

n

, and all

ℓ

\begin{matrix} (2.1) & 1 \leq | {C l}_{K} [ℓ] | \leq | {C l}_{K} | ≪_{n, ε} D_{K}^{1 / 2 + ε}, for all ε > 0 \end{matrix}

Our subject is a conjecture on the size of the

ℓ

-torsion subgroup, which suggests that (2.1) is far from the truth. We will focus primarily on cases when

ℓ

is prime, since

| {C l}_{K} [m] |

is multiplicative as a function of

m

, and for a prime

ℓ, | {C l}_{K} [ℓ^{t}] | \leq {| {C l}_{K} [ℓ] |}^{t}

(see [73]).

Conjecture 2.1 (

ℓ

-torsion conjecture). Fix a degree

n \geq 2

and a prime

ℓ

. Every number field

K / Q

of degree

n

satisfies

| {C l}_{K} [ℓ] | ≪_{n, ℓ, ε} D_{K}^{ε}

for all

ε > 0

This conjecture is due to Brumer and Silverman, in the more precise form: is it always true that

\log_{ℓ} | {C l}_{K} [ℓ] | ≪_{n, ℓ} \log D_{K} / \log \log D_{K} [17

, QUEstion

C l (ℓ, d)]

? Brumer and Silverman were motivated by counting elliptic curves of fixed conductor. Subsequently, this conjecture has appeared in many further contexts, including bounding the ranks of elliptic curves [34, $1.2]; bounding Selmer groups and ranks of hyperelliptic curves [10]; counting number fields [29, P. 166]; studying equidistribution of CM points on Shimura varieties [98, CONJECTURE 3.5]; and counting nonuniform lattices in semisimple Lie groups [6, THM. 7.5].

Conjecture 2.1 is known to be true for the degree

n = 2

and the prime

ℓ = 2

, when it follows from the genus theory of Gauss (see [68, ÑÐ½. 8.3]). This is the only case in which it is known. Nevertheless, starting in the early 2000s, significant progress has been made. The purpose of this survey is to give some insight into the wide variety of methods developed in recent work toward the conjecture. As an initial measure of progress, we define:

Property

C_{n, ℓ} (Δ)

. Fix a degree

n \geq 2

and a prime

ℓ

. Property

C_{n, ℓ} (Δ)

holds if for all number fields

K / Q

of degree

n, | {C l}_{K} [ℓ] | ≪_{n, ℓ, Δ, ε} D_{K}^{Δ + ε}

for all

ε > 0

Gauss proved that

C_{2, 2} (0)

holds. Until recently, no other case with

Δ < 1 / 2

was known.

The first progress was for imaginary quadratic fields. Suppose

K = Q (\sqrt{- d})

for a square-free integer

d > 1

, and suppose that

[a]

is a nontrivial element in

{C l}_{K} [ℓ]

for a prime

ℓ \geq 3

; thus

[a]

is the principal ideal class. Then by the Minkowski bound (1.1), there exists an integral ideal

b

[a]

such that

ℜ (b) ≪ d^{1 / 2}

. Moreover,

b^{ℓ}

is principal, say, generated by

(y + z \sqrt{- d}) / 2

for some integers

y, z

, and so

(ℜ (b))^{ℓ} = ℜ (b^{ℓ}) = (y^{2} + d z^{2}) / 4

. Consequently,

| {C l}_{K} [ℓ] |

can be dominated (up to a factor

d^{ε}

) by the number of integral solutions to

\begin{matrix} (2.2) & 4 x^{ℓ} = y^{2} + d z^{2}, with x ≪ d^{1 / 2}, y ≪ d^{ℓ / 4}, z ≪ d^{ℓ / 4 - 1 / 2} . \end{matrix}

When

ℓ = 3

, this can be interpreted in several ways: counting solutions to a congruence

y^{2} = 4 x^{3} (mod d)

; counting perfect square values of the polynomial

f (x, z) = 4 x^{3} - d z^{2}

; or counting integral points on a family of Mordell elliptic curves

y^{2} = 4 x^{3} - D

, with

D = d z^{2}

. Pierce used the first two perspectives, and Helfgott and Venkatesh used the third perspective, to prove for the first time that property

C_{2, 3} (Δ)

holds for some

Δ < 1 / 2 [48, 70

, 71]. (The Scholz reflection principle shows that

\log_{3} | {C l}_{Q (\sqrt{- d})} [3] |

and

\log_{3} | {C l}_{Q (\sqrt{3 d})} [3] |

differ by at most 1 , so results for 3-torsion apply comparably to both real and imaginary quadratic fields [76].) When

ℓ \geq 5

, the region in which

x, y, z

lie in (2.2) becomes inconveniently large relative to the trivial bound (2.1). Here is an open question: for a prime

ℓ \geq 5

, are there at most

≪ d^{Δ}

integral solutions to (2.2), for some

Δ < 1 / 2

Recently, Bhargava, Taniguchi, Thorne, Tsimerman, and Zhao made a breakthrough on property

C_{n, 2} (Δ)

for all

n \geq 3

. Fix a prime

ℓ

and a number field

K

of degree

n

. Given any nontrivial ideal class

[a] \in {C l}_{K} [ℓ]

, they show it contains an integral ideal

b

with

b^{ℓ}

a principal ideal generated by an element

β

lying in a well-proportioned "box." By an ingenious geometry of numbers argument, they show the number of such generators

β

in the box is

≪ D_{K}^{ℓ / 2 - 1 / 2}

. If

ℓ \geq 3

, this far exceeds the trivial bound (2.1), but if

ℓ = 2

, it slightly improves it. The striking refinement comes by recalling that any

β

of interest must also have

| N_{K / Q} (β) | = ℜ (b^{ℓ}) = (ℜ (b))^{ℓ}

be a perfect

ℓ

th power of an integer, say,

y^{ℓ}

. For

ℓ = 2

, they apply a celebrated result of Bombieri and Pila to count integral solutions

(x, y)

to the degree

n

equation

N_{K / Q} (β + x) = y^{2}

[15]. This strategy proves that property

C_{n, 2} (1 / 2 - 1 / 2 n)

holds for all degrees

n \geq 3

. Further refinements for degrees 3,4 show

C_{3, 2} (0.2785 \dots)

and

C_{4, 2} (0.2785 \dots)

hold; see [10].

Only two further nontrivial cases of property

C_{n, ℓ} (Δ)

are known, and for these we introduce the Ellenberg-Venkatesh criterion.

2.1. The Ellenberg-Venkatesh criterion

An important criterion for bounding

ℓ

-torsion in the class group of a number field

K

relies on counting small primes that are noninert in

K

. The germ of the idea, which has been credited independently to Soundararajan and Michel, goes as follows. Suppose, for example, that

K = Q (\sqrt{- d})

is an imaginary quadratic field with

d

square-free, and

ℓ

is an odd prime. Let

H

denote

{C l}_{K} [ℓ]

. Then

| H | = | {C l}_{K} | / [{C l}_{K} : H]

, and to show that

| H |

is small, it suffices to show that the index

[{C l}_{K} : H]

is large. Now suppose that

p_{1} \neq p_{2}

are rational primes not dividing

2 d

that both split in

K

, say,

p_{1} = p_{1} p_{1}^{σ}

and

p_{2} = p_{2} p_{2}^{σ}

,
where

σ

is the nontrivial automorphism of

K

. We claim that as long as

p_{1}, p_{2}

are sufficiently small,

p_{1}

and

p_{2}

must represent different cosets of

H

. Indeed, supposing to the contrary that

p_{1} H = p_{2} H

, one deduces that

p_{1} p_{2}^{σ} \in H

so that

{(p_{1} p_{2}^{σ})}^{ℓ}

is a principal ideal, say, generated by

(y + z \sqrt{- d}) / 2

, for some

y, z \in Z

. Taking norms shows

\begin{matrix} (2.3) & 4 {(p_{1} p_{2})}^{ℓ} = y^{2} + d z^{2} \end{matrix}

p_{1}, p_{2} < (1 / 4) d^{1 / (2 ℓ)}

, this forces

z = 0

, which yields a contradiction, since

4 {(p_{1} p_{2})}^{ℓ}

cannot be a perfect square. This proves the claim. In particular, if there are

M

such distinct primes

p_{1}, \dots, p_{M} < (1 / 4) d^{1 / 2 ℓ}

with

p_{j} ∤ 2 d

and

p_{j}

split in

K

, then

| {C l}_{K} [ℓ] | \leq | {C l}_{K} | M^{- 1}

Ellenberg and Venkatesh significantly generalized this strategy to prove an influential criterion, which we state in the case of extensions of

Q

[34]. (Throughout this survey, we will focus for simplicity on extensions of

Q

, but many of the theorems and questions we mention have analogues in the literature over any fixed number field.)

Ellenberg-Venkatesh criterion. Suppose

K / Q

is a number field of degree

n \geq 2

, fix an integer

ℓ \geq 2

, and fix

η < \frac{1}{2 ℓ (n - 1)}

. Suppose that there are

M

prime ideals

p_{1}, \dots, p_{M} \subset O_{K}

such that each

p_{j}

has norm

ℜ (p_{j}) < D_{K}^{η}

p_{j}

is unramified in

K

and

p_{j}

is not an extension of a prime ideal from any proper subfield of

K

. Then

\begin{matrix} (2.4) & | {C l}_{K} [ℓ] | ≪_{n, ℓ, ε} D_{K}^{\frac{1}{2} + ε} M^{- 1}, for all ε > 0 \end{matrix}

(A prime ideal

p \subset O_{K}

lying above a prime

p \in Q

is unramified in

K / Q

p^{2} ∤ p O_{K}

; a prime ideal

p \subset O_{K}

is an extension of a prime ideal in a proper subfield

K_{0} \subset K

if there exists a prime ideal

p_{0} \subset O_{K_{0}}

such that

p = p_{0} O_{K}

.) For example, if

p < D_{K}^{η}

is a rational prime that splits completely in

K

, so that

p O_{K} = p_{1} \dots p_{n}

for distinct prime ideals

p_{j}

, then each

p_{j}

satisfies the hypotheses of the criterion. In particular, if

M

rational (unramified) primes

p_{1}, \dots, p_{M} < D_{K}^{η}

split completely in

K

, then (2.4) holds. Alternatively, it suffices to exhibit prime ideals

p_{j} \subset O_{K}

of degree 1 , since such a prime ideal cannot be an extension of a prime ideal from a proper subfield.

Here is one of Ellenberg and Venkatesh's striking applications, which shows that

C_{2, 3} (1 / 3)

holds-the current record for

n = 2, ℓ = 3

. Fix a large square-free integer

d > 1

. Any prime

p ∤ 6 d

that is inert in

Q (\sqrt{- 3})

must split either in

Q (\sqrt{d})

or in

Q (\sqrt{- 3 d})

. Thus for any

η < 1 / 6

, at least one field

K \in {Q (\sqrt{d}), Q (\sqrt{- 3 d})}

has a positive proportion of the primes (1/2)

d^{η} \leq p \leq d^{η}

split in

K

. By the Ellenberg-Venkatesh criterion (2.4), this field

K

then has the property that

| {C l}_{K} [3] | ≪ D_{K}^{1 / 3 + ε}

for all

ε > 0

. By the Scholz reflection principle, this bound also applies to the other field in the pair, and

C_{2, 3} (1 / 3)

holds.

The Scholz reflection principle has also been generalized by Ellenberg and Venkatesh to bound

ℓ

-torsion (for odd primes

ℓ

) in class groups of even-degree extensions of certain number fields. In particular, by pairing their criterion with a reflection principle, they show that

C_{3, 3} (1 / 3)

holds and

C_{4, 3} (Δ)

holds for some

Δ < 1 / 2

[34, coR. 3.7]. This concludes the list of degrees

n

and primes

ℓ

for which property

C_{n, ℓ} (Δ)

is known for some

Δ < 1 / 2

Here are open problems: reduce the value

Δ < 1 / 2

for which

C_{n, ℓ} (Δ)

holds, when

n \geq 3

and

ℓ = 2

, or when

n = 2, 3

or 4 and

ℓ = 3

. For

n = 2, 3

or 4 and a prime

ℓ \geq 5

,
prove for the first time that

C_{n, ℓ} (Δ)

holds for some

Δ < 1 / 2

. For

n \geq 5

and a prime

ℓ \geq 3

, prove for the first time that

C_{n, ℓ} (Δ)

holds for some

Δ < 1 / 2

The Ellenberg-Venkatesh criterion underlies most of the significant recent progress on bounding

ℓ

-torsion in class groups. What is the best result it can imply? Assuming the Generalized Riemann Hypothesis, given any number field

K / Q

of degree

n

with

D_{K}

sufficiently large, a positive proportion of primes

p < D_{K}^{η}

split completely in

K

, implying

\begin{matrix} (2.5) & | {C l}_{K} [ℓ] | ≪_{n, ℓ, ε} D_{K}^{\frac{1}{2} - \frac{1}{2 ℓ (n - 1)} + ε}, for all ε > 0 \end{matrix}

As this is a useful benchmark, we will call this the GRH-bound, and for convenience set

Δ_{G R H} = \frac{1}{2} - \frac{1}{2 ℓ (n - 1)}

once

n, ℓ

have been fixed. Thus if GRH is true, for each

n, ℓ

, property

C_{n, ℓ} (Δ_{GRH})

holds. There has been intense interest in proving this without assuming GRH, and this will be our next topic.

3. FAMILIES OF FIELDS

So far we have considered, for each degree

n

, the "family" of number fields

K / Q

of degree

n

. Let us formalize this, letting

F_{n} (X)

be the set of all degree

n

extensions

K

Q

, with

D_{K} = | Disc (K / Q) | \leq X

; let

F_{n} = F_{n} (\infty)

. It is helpful at this point to consider more specific families of fields of a fixed degree. For example, we could define

F_{2}^{-} (X)

to be the set of imaginary quadratic fields

K

with

D_{K} \leq X

, and similarly

F_{2}^{+} (X)

for real quadratic fields. In general, given a transitive subgroup

G \subset S_{n}

, define the family

\begin{matrix} (3.1) & F_{n} (G; X) = {K / Q : \deg K / Q = n, Gal (\tilde{K} / Q) ≃ G, D_{K} \leq X} \end{matrix}

where all

K

are in a fixed algebraic closure

\bar{Q}, \tilde{K}

is the Galois closure of

K / Q

, the Galois group is considered as a permutation group on the

n

embeddings of

K

\bar{Q}

, and the isomorphism with

G

is one of permutation groups. When

F

is such a family, we define:

Property

C_{F, ℓ} (Δ)

holds if for all fields

K \in F, | {C l}_{K} [ℓ] | ≪_{n, ℓ, Δ, ε} D_{K}^{Δ + ε}

for all

ε > 0

Since Property

C_{F, ℓ} (Δ)

remains out of reach for almost all families, we also consider:

Property

C_{F, ℓ}^{*} (Δ)

holds if for almost all fields

K \in F, | {C l}_{K} [ℓ] | ≪_{n, ℓ, Δ, ε} D_{K}^{Δ + ε}

for all

ε > 0

. We say that a result holds for "almost all" fields in a family

F

if the subset

E (X)

of possible exceptions is density zero in

F (X)

, in the sense that

\frac{| E (X) |}{| F (X) |} \to 0 as X \to \infty

Here too, the first progress came for imaginary quadratic fields. Soundararajan observed that among imaginary quadratic fields with discriminant in a dyadic range

[- X, - 2 X]

, at most one can fail to satisfy

| {C l}_{K} [ℓ] | ≪ D_{K}^{1 / 2 - 1 / 2 ℓ + ε}

[82]. This verified

C_{F_{2}^{-}, ℓ}^{*} (Δ_{G R H})

for all primes

ℓ

. For

ℓ = 3

and quadratic fields, Wong observed that

C_{F_{2}^{\pm}, 3}^{*} (1 / 4)

holds [96]. For any odd prime

ℓ

, Heath-Brown and Pierce went below the GRH-bound, proving

C_{F_{2}^{-}, ℓ}^{*} (1 / 2 - 3 / (2 ℓ + 2))

[46]. They used the large sieve to show that aside from at most

O (X^{ε})

exceptions, all discriminants

- d \in [- X, - 2 X]

have

| {C l}_{Q (\sqrt{- d})} [ℓ] |

controlled by counting the number of distinct primes

p_{1}, p_{2}

of a certain size such that (2.3) has a nontrivial integral solution

(y, z)

. Then they showed there can be few such solutions, while averaging nontrivially over

d

. These methods relied heavily on the explicit nature of methods for imaginary quadratic fields. Fields of higher degree need a different approach.

3.1. Dual problems: counting primes, counting fields

To apply the Ellenberg-Venkatesh criterion, we face a question such as: "Given a field, how many small primes split completely in it?" This question is very difficult in general (and is related to the Generalized Riemann Hypothesis). There is a dual question: "Given a prime, in how many fields does it split completely?" Ellenberg, Pierce, and Wood devised a method to apply the Ellenberg-Venkatesh criterion by tackling the dual question instead [33]. The idea goes like this: suppose that each prime splits completely in a positive proportion of fields in a family

F

. Then the mean number of primes

p \leq x

that split completely in each field should be comparable to

π (x)

, and unless the primes conspire, almost all fields in

F

should have close to the mean number of primes split completely in them. To prove that the primes cannot conspire, Ellenberg, Pierce, and Wood developed a sieve method, modeled on the Chebyshev inequality from probability.

As input the sieve requires precise counts for the cardinality

N_{F} (X; p) =∣ {K \in F (X) : p splits completely in K} ∣

It also requires analogous counts

N_{F} (X; p, q)

for when two primes

p \neq q

split completely in

K

. Suppose one can prove that for some

σ > 0

and

τ < 1

, for all distinct primes

p, q

N_{F} (X; p, q) = δ (p q) | F (X) | + O ((p q)^{σ} | F (X) |^{τ})

for a multiplicative density function

δ (p q)

taking values in

(0, 1)

. Then Ellenberg, Pierce, and Wood prove that there exists

Δ_{0} > 0

(depending on

τ, σ

) such that the mean number of primes

p \leq X^{Δ_{0}}

that split completely in fields in

F (X)

is comparable to

π (X^{Δ_{0}})

. Moreover, there can be at most

O (| F (X) |^{1 - Δ_{0}})

exceptional fields

K

F (X)

such that fewer than half the mean number of primes split completely in

K

. Consequently, for any family

F

for which the crucial count (3.2) can be proved, combining this sieve with the Ellenberg-Venkatesh criterion proves that

C_{F, ℓ}^{*} (Δ)

holds for every integer

ℓ \geq 2

, where

Δ = max {\frac{1}{2} - Δ_{0}, Δ_{G R H}}

For which families of fields can (3.2) be proved? Counting number fields is itself a difficult question. For each integer

D \geq 1

, there are a finite number of extensions

K / Q

of degree

n

and discriminant exactly

D

, by Hermite's finiteness theorem [78, $4.1]. Let

N_{n} (X)

denote the number of degree

n

extensions

K / Q

with

D_{K} \leq X

(counted up to isomorphism). A folk conjecture, sometimes associated to Linnik, states that

N_{n} (X) \sim c_{n} X as X \to \infty

When

n = 2

, this is essentially equivalent to counting square-free integers (see [33, APPENDIX]). For degree

n = 3

, this is a deep result of Davenport and Heilbronn [28]. For degree

n = 4

, it is known by celebrated results of Cohen, Diaz y Diaz, and Olivier (counting quartic fields

K

with

Gal (\tilde{K} / Q) ≃ D_{4}

), and Bhargava (counting non-

D_{4}

quartic fields) [7,20]. For degree

n = 5

, it is known by landmark work of Bhargava [9].

The sieve method of Ellenberg, Pierce, and Wood requires an even more refined count (3.2), with prescribed local conditions and a power-saving error term with explicit dependence on

p, q

. Power saving error terms for

N_{n} (X)

were found for

n = 3

by Belabas, Bhargava, and Pomerance [5], Bhargava, Shankar, and Tsimerman [11], Taniguchi and Thorne [85]; for

n = 4

(non-

D_{4}

) by Belabas, Bhargava, and Pomerance [5]; and for

n = 5

by Shankar and Tsimerman [79]. These results can be refined to prove (3.2). Ellenberg, Pierce, and Wood used this strategy to prove that when

F

is the family of fields of degree

n = 2, 3, 4

(non-

D_{4}

), or

5, C_{F, ℓ}^{*} (Δ_{G R H})

holds for all sufficiently large primes

ℓ

. (For the few remaining small

ℓ

C_{F, ℓ}^{*} (Δ)

holds with a slightly larger

Δ < 1 / 2

.) Counting quartic

D_{4}

-fields with local conditions, ordered by discriminant, remains an interesting open problem.

The probabilistic method of Ellenberg-Pierce-Wood uses the property that the density function

δ (p q)

in (3.2) is multiplicative (i.e., local conditions at

p

and

q

are asymptotically independent). Frei and Widmer have adapted this approach to prove

C_{F, ℓ}^{*} (Δ_{G R H})

for all sufficiently large

ℓ

, for

F

a family of totally ramified cyclic extensions of

k

[40]. (That is,

F

comprises cyclic extensions

K / k

of degree

n

in which every prime ideal of

O_{k}

not dividing

n

is either unramified or totally ramified in

K

). This family is chosen since the density function

δ (p q)

is multiplicative. It would be interesting to investigate whether a probabilistic method can rely less strictly upon multiplicativity of the density function.

There is a great obstacle to expanding the above approach to the family of all fields of degree

n

when

n \geq 6

. Then, even the asymptotic (3.3) is not known. For each

n \geq 6

\begin{matrix} (3.4) & N_{n} (X) \leq a_{n} X^{c_{0} (\log n)^{2}} \end{matrix}

is the best-known bound, with

c_{0} = 1.564

, by Lemke Oliver and Thorne [61]; this improves on Couveignes [25], Ellenberg and Venkatesh [36], and Schmidt [75]. For lower bounds, in general the record is

N_{n} (X) ≫ X^{1 / 2 + 1 / n}

, for all

n \geq 7

[12]. For any

n

divisible by

p = 2, 3

or 5, KlÃ¼ners (personal communication) has observed that

N_{n} (X) ≫ X

, since there exists a field

F / Q

of degree

n / p

such that degree

p S_{p}

-extensions of

F

exhibit linear asymptotics

Tackling the problem of counting primes with certain splitting conditions in a specific field via the dual problem of counting fields with certain local conditions at specific primes seems out of reach for higher degree fields. How about tackling the problem of counting primes directly?

4. COUNTING PRIMES WITH L-FUNCTIONS

The prime number theorem states that the number

π (x)

of primes

p \leq x

satisfies

π (x) \sim Li (x)

x \to \infty

. To count small primes, or primes in short intervals, requires understanding the error term, as well as the main term. For each

1 / 2 \leq Δ < 1

, the statement

\begin{matrix} (1.1) & π (x) = Li (x) + O (x^{Δ + ε}) for all ε > 0 \end{matrix}

is essentially equivalent to the statement that the Riemann zeta function

ζ (s)

is zero-free for

ℜ (s) > Δ

[27, cÐ½. 18]. The Riemann Hypothesis conjectures this is true for

Δ = 1 / 2

; it is
not known for any

Δ < 1

. The best known Vinogradov-Korobov zero-free region is:

\begin{matrix} (4.2) & σ \geq 1 - \frac{C}{(\log t)^{2 / 3} (\log \log t)^{1 / 3}}, t \geq 3 \end{matrix}

with an absolute constant

C > 0

computed by Ford [37].

To count primes with a specified splitting type in a Galois extension

L / Q

of degree

n_{L} \geq 2

, consider the counting function

\begin{matrix} (4.3) & π_{C} (x, L / Q) = | {p \leq x : p unramified in L, [\frac{L / Q}{p}] = C} | \end{matrix}

in which

[\frac{L / Q}{p}]

is the Artin symbol and

C

is any fixed conjugacy class in

G = Gal (L / Q)

. For example, when

L = Q (e^{2 π i / q})

, this can be used to count primes in a fixed residue class modulo

q

. Or, for example, for any Galois extension

L / Q

, when

C = {I d}

, this counts primes that split completely in

L

. By the celebrated Chebotarev density theorem [88],

\begin{matrix} (4.4) & π_{C} (x, L / Q) \sim \frac{| C |}{| G |} Li (x), as x \to \infty \end{matrix}

But just as for

π (x)

, to count small primes accurately requires more quantitative information. A central goal is to prove an asymptotic for

π_{C} (x, L / Q)

that is valid for

x

very small relative to

D_{L} = | Disc L / Q |

, and with an effective error term. This requires exhibiting a zero-free region for the Dedekind zeta function

ζ_{L} (s)

. This is more complicated than (4.2), due to the possibility of an exceptional Landau-Siegel zero: within the region

\begin{matrix} (4.5) & σ \geq 1 - {(4 \log D_{L})}^{- 1}, | t | \leq {(4 \log D_{L})}^{- 1} \end{matrix}

ζ_{L} (σ + i t)

can contain at most one (real, simple) zero, denoted

β_{0}

if it exists. (As observed by Heilbronn and generalized by Stark, if

β_{0}

exists then it must "come from" a quadratic field, in the sense that

L

contains a quadratic subfield

F

with

ζ_{F} (β_{0}) = 0 [47, 83]

Lagarias and Odlyzko used the zero-free region (4.5) to prove there exist absolute, computable constants

C_{1}, C_{2}

such that for all

x \geq \exp (10 n_{L} {(\log D_{L})}^{2})

\begin{matrix} (4.6) & | π_{C} (x, L / Q) - \frac{| C |}{| G |} Li (x) | \leq \frac{| C |}{| G |} Li (x^{β_{0}}) + C_{1} x \exp (- C_{2} n_{L}^{- 1 / 2} (\log x)^{1 / 2}) \end{matrix}

in which the

β_{0}

term is present only if

β_{0}

exists (see [60], and Serre [77]). This was the first effective Chebotarev density theorem. It can be difficult to apply to questions of interest because of the mysterious

β_{0}

term, and because

x

must be a large power of

D_{L}

(certainly at least

x \geq D_{L}^{10 n_{L}}

). In contrast, to apply the Ellenberg-Venkatesh criterion to a field

K

of degree

n

, we aim to exhibit primes

p < D_{K}^{η}

that split completely in the Galois closure

\tilde{K}

(and hence in

K

), with

η \approx 1 / (2 ℓ (n - 1)) \to 0

n, ℓ \to \infty

. (These primes are even smaller relative to

D_{\tilde{K}}

, since

D_{K}^{| G | / n} ≪_{G} D_{\tilde{K}} ≪_{G} D_{K}^{| G | / 2}

, where

G = Gal (\tilde{K} / Q)

[72].)

If GRH holds for

ζ_{L} (s)

, then

ζ_{L} (s)

is zero-free for

ℜ (s) > 1 / 2

, and Lagarias and Odlyzko improve (4.6) in three ways: (i) it is valid for

x \geq 2

; (ii) the

β_{0}

term is not present; (iii) the remaining error term is

O (x^{1 / 2} \log (D_{L} x^{n_{L}}))

. Properties (i) and (ii) show that for every

η > 0

, for every degree

n

extension

K / Q

with

D_{K}

sufficiently large, at least

≫ π (D_{K}^{η})

primes

p \leq D_{K}^{η}

split completely in the Galois closure

\tilde{K}

(and hence in

K

). When input into
the Ellenberg-Venkatesh criterion, this is the source of the GRH-bound (2.5) for all integers

ℓ \geq 2

Here is a central goal: improve the Chebotarev density theorem (4.6) without assuming GRH, so that (i') for any

η > 0

it is valid for

x

as small as

x \geq D_{L}^{η}

(for all

D_{L}

sufficiently large) and (ii) the

β_{0}

term is not present. (For many applications, the final error term in (4.6) suffices as is.) If this held for

L = \tilde{K}

the Galois closure of a field

K

, the Ellenberg-Venkatesh criterion would imply the GRH-bound (2.5) for

ℓ

-torsion in

{C l}_{K}

for all integers

ℓ \geq 2

, without assuming GRH. Recently, Pierce, Turnage-Butterbaugh, and Wood showed that the key improvements (i') and (ii) hold if for some

0 < δ \leq 1 / 4, ζ_{L} (s) / ζ (s)

is zero-free for

s = σ + i t

in the box

\begin{matrix} (4.7) & 1 - δ \leq σ \leq 1, | t | \leq \log D_{L}^{2 / δ} \end{matrix}

Proving this for any particular

L

-function

ζ_{L} (s) / ζ (s)

of interest is out of reach. Instead, it can be productive to study a family of

L

-functions. In particular, if

F = F_{n} (G; X)

is a family of degree

n

fields with fixed Galois group of the Galois closure, property

C_{F, ℓ}^{*} (Δ_{G R H})

will follow (for all integers

ℓ \geq 2

) if it is true for almost all fields

K \in F_{n} (G; X)

, that

ζ_{\tilde{K}} (s) / ζ (s)

is zero-free in the box (4.7). This was the strategy Pierce, Turnage-Butterbaugh, and Wood developed in [72], which we will now briefly sketch.

4.1. Families of $L$ -functions

There is a long history of estimating the density of zeroes within a certain region, for a family of

L

-functions. If we can show there are fewer possible zeroes in the region than there are

L

-functions in the family, then some of the

L

-functions must be zero-free in that region. We single out a result of Kowalski and Michel, who used the large sieve to prove a zero density result for families of cuspidal automorphic

L

-functions [56]. In particular, for suitable families, their result implies that almost all

L

-functions in the family must be zero-free in a box analogous to (4.7).

There are two fundamental barriers to applying this to our problem of interest: the representation underlying

ζ_{\tilde{K}} (s) / ζ (s)

is not always cuspidal, and it is not always known to be automorphic. Suppose

G

has irreducible complex representations

ρ_{0}, ρ_{1}, \dots, ρ_{r}

, with

ρ_{0}

the trivial representation. Then for

K \in F_{n} (G; X), ζ_{\tilde{K}}

is a product of Artin

L

-functions,

\begin{matrix} (4.8) & ζ_{\tilde{K}} (s) / ζ (s) = \prod_{j = 1}^{r} L {(s, ρ_{j}, \tilde{K} / Q)}^{\dim ρ_{j}} \end{matrix}

The Artin (holomorphy) conjecture posits that for each nontrivial irreducible representation

ρ_{j}, L (s, ρ_{j}, \tilde{K} / Q)

is entire. The (strong) Artin conjecture posits that for each nontrivial irreducible representation

ρ_{j}

, there is an associated cuspidal automorphic representation

π_{\tilde{K}, j}

G L (m_{j}) / Q

, and

L (s, π_{\tilde{K}, j}) = L (s, ρ_{j}, \tilde{K} / Q)

. This is known for certain types of representations of certain groups, but otherwise is a deep open problem (see recent work in [19]). For the moment, we will proceed by assuming the strong conjecture. Then the factorization (4.8) naturally slices the family

ζ_{\tilde{K}} (s) / ζ (s)

, as

K

varies over

F_{n} (G; X)

, into

r

families

L_{1} (X), L_{2} (X), \dots, L_{r} (X)

, where each

L_{j} (X)

is the set of cuspidal automorphic representations

π_{\tilde{K}, j}

associated to the representation

ρ_{j}

. Kowalski and Michel's result applies
to each family

L_{j} (X)

individually. This proves that every representation

π \in L_{j} (X)

has associated

L

-function

L (s, π)

being zero-free in the box (4.7)â€”except for a possible subset of "bad" representations

π

, of density zero in

L_{j} (X)

, for which

L (s, π)

could have a zero in the box. (Of course, no such zero exists if GRH is true, but we are not assuming GRH.)

Now a crucial difficulty arises: if there were a "bad" representation

π \in L_{j} (X)

, in how many products (4.8) could it appear, as

K

varies over

F_{n} (G; X)

? Each field

K

for which the "bad" factor

L (s, π)

appears could have a zero of

ζ_{\tilde{K}} (s) / ζ (s)

in (4.7). Thus the crucial question is: for a fixed nontrivial irreducible representation

ρ

G

, how many fields

K_{1}, K_{2} \in F_{n} (G; X)

have

L (s, ρ, {\tilde{K}}_{1} / Q) = L (s, ρ, {\tilde{K}}_{2} / Q)

? This can be stated a different way. Given a subgroup

H

G

, let

{\tilde{K}}^{H}

denote the subfield of

\tilde{K}

fixed by

H

. It turns out that the question can be transformed into: how many fields

K_{1}, K_{2} \in F_{n} (G; X)

have

{\tilde{K}}_{1}^{Ker (ρ)} = {\tilde{K}}_{2}^{Ker (ρ)}

? Let us call this a collision. If a positive proportion of fields in

F_{n} (G; X)

can collide for

ρ_{j}

, then via the factorization (4.8), the possible existence of even one "bad" element in

L_{j} (X)

could allow a positive proportion of the functions

ζ_{\tilde{K}} (s) / ζ (s)

to have a zero in (4.7). In particular, then this approach would fail to prove

C_{F, ℓ}^{*} (Δ_{G R H})

for the family

F = F_{n} (G; X)

. To rule this out, we aim to show that for each nontrivial irreducible representation

ρ_{j}

G

, collisions are rare.

We define the "collision problem" for the family

F_{n} (G; X)

: how big is

\begin{matrix} (4.9) & max_{ρ} max_{K_{1} \in F_{n} (G; X)} | {K_{2} \in F_{n} (G; X) : {\tilde{K}}_{1}^{Ker (ρ)} = {\tilde{K}}_{2}^{Ker (ρ)}} | ? \end{matrix}

Here the maximum is over the nontrivial irreducible representations

ρ

G

with

Ker (ρ)

a proper normal subgroup of

G

. Suppose for a particular family

F_{n} (G; X)

, the collisions (4.9) number at most

≪ X^{α}

. Then the strategy sketched here ultimately shows that aside from at most

≪ X^{α + ε}

exceptional fields (for any

ε > 0

), every field in

K \in F_{n} (G; X)

has the property that an improved Chebotarev density theorem with properties (i') and (ii) holds for its Galois closure

\tilde{K}

. If we can prove simultaneously that

| F_{n} (G; X) | ≫ X^{β}

for some

β > α

, then the improved Chebotarev density theorem holds for almost all fields in the family. Consequently, we would obtain property

C_{F, ℓ}^{*} (Δ_{G R H})

, for all integers

ℓ \geq 2

Thus the goal of bounding

ℓ

-torsion in class groups of fields in the family

F_{n} (G; X)

has been transformed into a question of counting how often certain fields share a subfield. For which families can the collision problem (4.9) be controlled? For some groups, the number of collisions can be

≫ | F_{n} (G; X) |

(for example,

G = Z / 4 Z)

. On the other hand, if

G

is a simple group, or if all nontrivial irreducible representations of

G

are faithful, the number of collisions is

≪ 1

(but a lower bound

| F_{n} (G; X) | ≫ X^{β}

for some

β > 0

may not be known, yet). In general, controlling the collision problem is difficult.

One idea is to restrict attention to an advantageously chosen subfamily of fields, call it

F_{n}^{*} (G; X) \subset F_{n} (G; X)

. To bound (4.9) within a subfamily it suffices to count

\begin{matrix} (4.10) & max_{H} max_{\deg (F / Q) = [G : H]} | {K \in F_{n}^{*} (G; X) : {\tilde{K}}^{H} = F} | \end{matrix}

Here

H

ranges over the proper normal subgroups of

G

that appear as the kernel of some nontrivial irreducible representation. For some groups

G

, if

F_{n}^{*} (G; X)

is defined appropriately, this can be further transformed into counting number fields with fixed discriminant.

Let us see how this goes in the example

G = S_{n}

with

n = 3

n \geq 5

, so that

A_{n}

is the only nontrivial proper normal subgroup (the kernel of the sign representation). Consider the subfamily

F_{n}^{*} (S_{n}; X)

of fields with square-free discriminant. (These are a positive proportion of all degree

n S_{n}

-fields for

n \leq 5

and conjecturally so for

n \geq 6

.) Then for

H = A_{n}

and

F

a fixed quadratic field, it can be shown that any field

K

counted in (4.10) must have the property that

D_{K} = D_{F}

(up to some easily controlled behavior of wildly ramified primes). Under this very strong identity of discriminants, (4.10) is dominated by

\begin{matrix} (4.11) & max_{D \geq 1} | {K \in F_{n}^{*} (S_{n}; X) : D_{K} = D} | \end{matrix}

This strategy transforms the collision problem into counting fields of fixed discriminant.

For certain other groups

G

, (4.10) can also be dominated by a quantity analogous to (4.11) if the subfamily

F_{n}^{*} (G; X)

is defined by specifying that each prime that is tamely ramified in

K

has its inertia group generated by an element in a carefully chosen conjugacy class

I

G

. For such a group

G

, the final step in this strategy for proving improved Chebotarev density theorems for almost all fields in the family

F_{n}^{*} (G; X)

is to bound (4.11). If

| F_{n}^{*} (G; X) | ≫ X^{β}

is known, it suffices to prove (4.11) is

≪ X^{α}

for some

α < β

. In general, counting number fields with fixed discriminant is very difficult-we will return to this problem later. But for some families

F_{n}^{*} (G; X)

, (4.11) can be controlled sufficiently well, relative to a known lower bound for

| F_{n}^{*} (G; X) |

This is the strategy developed by Pierce, Turnage-Butterbaugh, and Wood in [72]. The result is an improved Chebotarev density theorem, with properties (i') and (ii), that holds unconditionally for almost all fields in the following families: (a)

F_{p} (C_{p}; X)

cyclic extensions of any prime degree; (b)

F_{n}^{*} (C_{n}; X)

totally ramified cyclic extensions of any degree

n \geq 2

; (c)

F_{p}^{*} (D_{p}; X)

prime degree dihedral extensions,

I

being the class of order 2 elements; (d)

F_{n}^{*} (S_{n}; X)

fields of square-free discriminant,

n = 3

, 4 ; and (e)

F_{4}^{*} (A_{4}; X)

I

being either class of order 3 elements. Conditional on the strong Artin conjecture, they proved the improved Chebotarev density theorem also holds for almost all fields in the following families: (f)

F_{5}^{*} (S_{5}; X)

quintic fields of square-free discriminant; and (g)

F_{n} (A_{n}; X)

, for all

n \geq 5

. (There are other families, such as

F_{n}^{*} (S_{n}; X)

for

n \geq 6

, to which the strategy applies, but the current upper bound known for (4.11) is larger than the known lower bound for

| F_{n}^{*} (S_{n}; X) |

.) As a consequence, Pierce, Turnage-Butterbaugh, and Wood proved for each family (a)-(e) that

C_{F, n}^{*} (Δ_{GRH})

holds unconditionally for all integers

ℓ \geq 2

, and it holds for each family (f)-(g) under the strong Artin conjecture. This was the first time such a result was proved for families of fields of arbitrarily large degree.

4.2. Further developments

Since the work outlined above, many interesting new developments have followed, relating to zero density results for families of

L

-functions, Chebotarev density theorems for families of fields, and

ℓ

-torsion in class groups of fields in specific families.

First, there has been renewed interest in zero density results for families of

L

-functions, concerning potential zeroes in regions close to the line

ℜ (s) = 1

, and extending the perspective of Kowalski and Michel [56]; see, for example, [18,49,87].

Second, several new strategies have focused on the problem of proving effective Chebotarev density theorems for almost all fields in a family. The work in [72] raised several desiderata. Some groups

G

have the property that no ramification restriction exists that allows the "collision problem" in the form (4.10) to be transformed into a "discriminant multiplicity problem" in the form (4.11). For example, this occurs for any noncyclic abelian group, or

D_{4}

. These cases remain open; instead, An recently proved a Chebotarev density theorem for almost all fields in a family of quartic

D_{4}

-fields associated to a fixed biquadratic field [2]. Another significant desideratum was to remove the dependence on the strong Artin conjecture. Thorner and Zaman recently achieved this, by proving a zero density estimate directly for Dedekind zeta functions, without passing through the factorization (4.8) [86]. But that work is still explicitly conditional on the ability to control a collision problem similar to (4.9), for which the best known strategy is still the approach of [72].

Most recently, the collision problem has been bypassed for certain groups

G

by interesting new work of Lemke Oliver, Thorner, and Zaman [62]. Their key idea when studying fields in a family

F_{n} (G; X)

is to prove a zero-free region not for

ζ_{\tilde{K}} / ζ

but for

ζ_{\tilde{K}} / ζ_{{\tilde{K}}^{N}}

where

N

is a nontrivial normal subgroup of

G

. This allows them to replace a collision problem like (4.9) by an "intersection multiplicity problem," bounding

\begin{matrix} (4.12) & max_{K_{1} \in F_{n} (G; X)} | {K_{2} \in F_{n} (G; X) : {\tilde{K}}_{1} \cap {\tilde{K}}_{2} \neq {\tilde{K}}_{1}^{N} \cap {\tilde{K}}_{2}^{N}} | \end{matrix}

The number of exceptional fields, for which a desired Chebotarev-type theorem cannot be verified, is then dominated by (4.12) (up to

X^{ε}

). This is advantageous if

G

has a unique minimal nontrivial normal subgroup

N

, so that (4.12) is

≪ 1

. But as a trade-off, one no longer obtains an effective Chebotarev density theorem for each conjugacy class

C

G

Let

π_{K} (x)

count prime ideals

p \subset O_{K}

with

ℜ_{K / Q} p \leq x

. Let

F

represent either of the two following families: degree

p

fields

K / Q

for

p

prime, or degree

n S_{n}

-fields

K / Q

, for any

n \geq 2

. Lemke Oliver, Thorner, and Zaman prove that except for at most

≪ X^{ε}

exceptional fields, every

K \in F (X)

has

| π_{K} (x) - π (x) | \leq C_{1} x \exp (- C_{2} \sqrt{\log x})

for every

x \geq {(\log D_{K})}^{C_{3} (n, ε)}

. In either family

F

, they obtain results on

ℓ

-torsion by applying the Ellenberg-Venkatesh criterion using prime ideals of degree 1 . If

π_{K}^{*} (x)

counts only prime ideals of degree 1 , then

π_{K}^{*} (x) = π_{K} (x) + O_{n} (\sqrt{x})

, so the above result exhibits many small prime ideals of degree 1 . Thus for either family,

C_{F, n}^{*} (Δ_{G R H})

holds unconditionally for all

ℓ

(and the exceptional set is very small). (They also exhibit infinitely many degree

n S_{n}

-fields

K

with

{C l}_{K}

as large as possible, but

| {C l}_{K} [ℓ] |

3.3. Generalizing the ErdÃ¶s-Vaaler argument

3.4. An iterative compression algorithm

3.5. The quality increment argument

3.6. Proof of Theorem 3.5

ACKNOWLEDGMENTS

FUNDING

REFERENCES

DIMITRIS KOUKOULOPOULOS

EULER SYSTEMS AND THE BLOCH-KATO CONJECTURE FOR AUTOMORPHIC GALOIS REPRESENTATIONS

DAVID LOEFFLER AND SARAH LIVIA ZERBES

Abstract

MATHEMATICS SUBJECT CLASSIFICATION 2020

KEYWORDS

1. WHAT IS THE BLOCH-KATO CONJECTURE?

Critical values

Iwasawa theory

2. WHAT IS AN EULER SYSTEM?

3. EULER SYSTEMS FOR SHIMURA VARIETIES

Choosing the data

4. DEFORMATION TO CRITICAL VALUES

5. CONSTRUCTING p p ppp-ADIC L L LLL-FUNCTIONS

Coherent cohomology

Interpolation

6. P-ADIC REGULATORS

7. DEFORMATION TO CRITICAL VALUES

Non-regular weights

REFERENCES

DAVID LOEFFLER

SARAH LIVIA ZERBES

COUNTING PROBLEMS: CLASS GROUPS, PRIMES, AND NUMBER FIELDS

LILLIAN B. PIERCE

ABSTRACT

MATHEMATICS SUBJECT CLASSIFICATION 2020

KEYWORDS

1. HISTORICAL PRELUDE

1.1. The class group

2. THE ℓ â„“ â„“\ellâ„“-TORSION CONJECTURE

2.1. The Ellenberg-Venkatesh criterion

3. FAMILIES OF FIELDS

3.1. Dual problems: counting primes, counting fields

4. COUNTING PRIMES WITH L-FUNCTIONS

4.1. Families of L L LLL-functions

4.2. Further developments

5. CONSTRUCTING $p$ -ADIC $L$ -FUNCTIONS

2. THE $ℓ$ -TORSION CONJECTURE

4.1. Families of $L$ -functions